Patent Publication Number: US-2021182298-A1

Title: Gait-based biometric data analysis system

Description:
RELATED APPLICATIONS 
     This application is a Continuation-in-Part of U.S. application Ser. No. 15/826,744 filed on Nov. 30, 2017, which is a Divisional of U.S. Pat. No. 9,864,780 filed Nov. 19, 2015, which is a Continuation-in-Part of U.S. Pat. No. 9,204,797 filed Jul. 11, 2013, which is a Continuation-in-Part of U.S. Pat. No. 9,188,963 filed Dec. 6, 2012. 
    
    
     TECHNICAL FIELD 
     The present invention relates to a gait based biometric data analysis system used for the detection and isolation of movement and mobility biomarkers and which can be used for the determination of the progression or regression of health, wellness, and fitness. 
     BACKGROUND 
     The increase in the use of remote patient monitoring, mobile healthcare, and user discrete analytical fields have highlighted some shortcomings of current personal and population-based analytics systems as well as the effects of generally prescribed pharmacological and therapeutic treatments after diagnosis. 
     Analytical systems generally come in a number of categories. Data, text, and imaging systems use large installed or network-based hardware and software platforms which, when used, diagnoses the user&#39;s condition using well-known techniques and methods. Less cumbersome and intrusive analytical systems and methods include question and answers done orally or written, which measure temperature, weight, pulse rate, as well as a plethora of other biological and physiological indicators, including through the experienced eye of a practitioner, clinician, therapist or surgeon specialist. These indicators are used when searching for signs of health, fitness level, and disease. The ancient medicinal art of reflexology is one historical effort focused on the correlation between disease pathology and the feet. Adverse reactions to prescribed pharmacological and therapeutic treatments of disease are known to adversely affect a person&#39;s stride, balance, weight, joints, tendons, and ligaments and, as a consequence, that person&#39;s gait as well. 
     The above noted analytical systems have their drawbacks. Specifically, current techniques may involve extremely large and use controlled or proprietary software algorithms. They are also fundamentally reliant on the quality and efficacy of the data input. Similarly, question and answers obtained orally or in written form are area specific, limited in scope and use, and require more active participation and knowledge from the user. In addition, current techniques are such that the experienced eye of the practitioner, clinician, therapist or surgeon only lasts as long as a patient&#39;s visit to the clinic/health facility. These and other current systems have been seen as too invasive, too cumbersome for some people to use, limited in scope, limited in reliability and utility of data, or simply too complicated to understand. 
     There is therefore a need for an analytical system that is neither invasive nor linear in scope and use and which provides access to quality and reliable data gathered by convenient user-worn devices. 
     SUMMARY 
     The present invention provides systems and methods for determining a user&#39;s health/wellness condition. Gait-based biometric data from a user is gathered using a sensor module. The biometric data is transmitted to a data processing module that compares characteristics of the biometric data with previously obtained baseline biometric data from the same user. Differences between the current data and the baseline data indicate changes in the user&#39;s condition. Databases containing kinematic chain models for the user are used to obtain more accurate and more specific indications regarding determined changes in the user&#39;s condition. A base kinematic chain model is created for the user when the user first uses the system and current kinematic chain models are generated for each biometric data set gathered. Characteristics of the base data and the current kinematic chain models are compared to determine changes in the user&#39;s condition. 
     In a first aspect, the present invention provides a system for determining a change in a user&#39;s condition, the system comprising:
         at least one sensor module comprising at least one sensor for gathering gait-based biometric data from said user, said sensor module being in a single device having at least two sensors;   a data storage module for storing data relating to baseline data, said baseline data being derived from said gait-based biometric data gathered from said at least one sensor module when said user first uses said system;   a data processing module for receiving data from said sensor module, said data processing module being for comparing characteristics of said baseline data with characteristics of said data received from said sensor module;   at least one database in communication with said data processing module, said at least one database containing data relating to a base kinematic chain model specific to said user, said base kinematic chain model being derived from said baseline data;       

     wherein
         said data processing module derives a current kinematic chain model from said data received from said at least one sensor module;   said data processing module compares characteristics of said current kinematic chain model with characteristics of said base kinematic chain model;   a change in said user&#39;s condition is indicated when said characteristics of said data received from said at least one sensor module are not within predetermined limits of said characteristics of said baseline data.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The embodiments of the present invention will now be described by reference to the following figures, in which identical reference numerals in different figures indicate identical elements and in which: 
         FIG. 1  is a block diagram of the system according to one aspect of the invention; 
         FIG. 2A  is an image illustrating the different forces applied by a human foot as it takes a step; 
         FIG. 2B  is a diagram illustrating different zones on an insole with at least one sensor placement per zone according to one embodiment of another aspect of the invention; 
         FIG. 3  illustrates raw data waveforms and data waveforms after a low pass filter has been applied; 
         FIG. 4  illustrates loops plotted using raw data and filtered data; 
         FIG. 5  illustrates a number of characteristics for different sets of data from the same user, as well as an average characteristic loop derived from the other loops; 
         FIG. 6  illustrates the different characteristics which may be derived from the characteristic loops; 
         FIG. 7  illustrates characteristic loops using highly correlated data; 
         FIG. 8  shows average characteristic loops for different users; 
         FIG. 9  is a block diagram illustrating the data connections between various components of the system according to another aspect of the invention; 
         FIG. 10  is a flowchart illustrating the steps in a method according to another aspect of the invention; and 
         FIG. 11  is an illustration of an individual user&#39;s baseline loop signature for a left foot and a right foot. 
     
    
    
     DETAILED DESCRIPTION 
     In one aspect, the present invention can be used for the detection of known pathological conditions associated with disease states and for the detection of the effects of prescribed pharmacological and therapeutic treatments. 
     In another aspect, the present invention provides analytical systems and methods for the assessment of movement, balance, weight and mobility based on user gait. As well, the present invention allows for intelligent pattern matching to assess kinematic chain models spanning all 26 bones and 31 joints in each foot and ankle, up through the lower leg, upper leg, pelvis, sacral vertebrae, and lumbar vertebrae. These kinematic chain models describe nominal gait, supination and pronation, and excessive supination and excessive pronation of the foot. In addition, research has demonstrated shared disease patterns useful for the detection of diabetes, Alzheimer&#39;s, and Parkinson&#39;s disease among other ailments, sicknesses, and injuries. Each of these patterns has a direct correlation to a range of sport injuries conditions such as concussion, head trauma, as well as neurological ailments and orthopedics. A sensor module with multiple sensors is placed inside a user&#39;s shoe and biometric data is gathered from the sensors when the user takes a step, walks, runs, jumps, dances, or shuffles. The data extracted from regions using groups of pressure sensors, groups of thermal sensors, and other sensor types is used to generate loops as the various sets of data are plotted against each other to form a loop-based biometric of a user. The loops generated from the data are then compared against stored loops previously obtained from known individuals having nominal function or specific identified diseases, injury or sickness and loop data recorded and annotated to indicate specific pathologies associated with the progression or regression of the disease, sickness or injury state, as well as other characteristic data extrapolated from the loops. Based on the results of the comparison, the user&#39;s movement, weight and other mobility characteristics are assessed using predetermined indicators in conjunction with analytical input from distributed databases and with the input of each user&#39;s specific characteristic data from which other data can be extrapolated. 
     It should be clear that, using the biometric data, it can be determined whether the user has increased activity and performance of basic movement functions, i.e., shuffling, walking, running, dancing, jumping, or whether the user has the proper fitting insole insert and, if not, recommendations can be made. As a result of this analysis, recommendations can be made to the user with the recommendations being focused on directing changes in their gait to thereby lead to a more optimal kinematic skeletal performance. Similarly, the data gathered can be used to determine whether the footwear the user is using is of a proper fit or whether recommendations for alternative footwear types and models is warranted. As well, the data and the system can be used to determine whether the user has a specific condition or ailment, whether a specific condition or ailment is worsening, or whether a specific condition or ailment is improving. These results can be presented for use by attending physicians as disease or injury detection indicators, or subsequently, as disease or injury progression indicators. 
     In one embodiment, the user&#39;s biometric data is, preferably, previously extracted from his gait. The previously gathered data, and the plotted loops derived therefrom, can be used as a baseline for the user. Subsequent biometric data sets gathered from the user can then be compared against the baseline. Depending on the comparison results, a range of analyses can be performed to determine changes in the user&#39;s condition. Such changes can include progression or regression of a user&#39;s movement and/or mobility. Data gathered from the general population can be used to establish any correlation between a person&#39;s changing gait as he or she progresses or regresses in a specific health and fitness condition. Treatment effects of prescribed pharmacological and/or therapeutic remedies can also be determined using the baseline biometric data from the user as the user progresses in his or her daily programs and treatment. Periodic gathering of the user&#39;s biometric data can be used to track and monitor the effects of the treatment on the user&#39;s gait to establish any causal link between the treatment regime and the user&#39;s gait and other gait-based characteristics such as weight and load bearing weight. Such links and the specific effects of the fitness program or treatment regime can then be used to further heighten the effectiveness of shoe-based biometric data gathering devices as diagnostic tools. In addition to the above, the diagnostic tools can be supplemented by a network of distributed databases that have pathomechanical movement and mobility data including kinematic models of the human skeleton spanning all 26 bones and 31 joints in each foot and ankle, up through the lower leg, upper leg, pelvis, sacral vertebrae, and lumbar vertebrae. Such data, in conjunction with data received from an insole and with a user&#39;s specific characteristics, can be used to narrow down a suitable diagnosis for the user&#39;s pathomechanical abnormality. Such network of distributed databases storing a population of movement and mobility data and related characteristic data such as foot shape, foot type, standing weight and posture can be used to narrow down gait-based movement and mobility biomarkers associated with a range of performance, human ailments, injury and disease. 
     To assist in the implementation of the present invention, kinematic models of the human skeleton have been formed as constructs. In the present invention, the kinematic models are defined using Denavit-Hartenberg conventions, and the dynamic models are defined using the Jacobian conventions. 
     For the models used in the present invention, geometry is applied to the analysis of the movement of multi-degree of freedom kinematic chains that form the structure of skeletal systems. The emphasis on geometry means that the links are modeled as rigid bodies and joints are assumed to provide pure rotation or translation. 
     The kinematics models used within the present invention can be stored within a network of distributed databases and these models map the relationship between the dimensions and connectivity of kinematic chains and the position, velocity and acceleration of each of the links in the skeletal system in order to assess and approximate movement and to interpret forces and torques. The relationship between mass and inertia properties, motion, and the associated forces and torques may be analyzed using the network of distributed databases as part of skeletal dynamics. 
     A further aspect of the systems according to the present invention encodes established pathomechanical kinematic models specific to disease pathologies, injury and sickness. These models are stored and used within a network of distributed databases and can be used in the comparison of characteristic loop data input from the gait-based biometric device and the kinematic data models stored within the network of distributed databases. 
     The present invention also provides systems and methods that relate to a mathematical formalism that accurately specifies the position of each joint, the velocity of each joint, and the acceleration of each joint. When a joint is prismatic, then the velocity is a linear velocity and the acceleration is a linear acceleration. When a joint is rotational, then the velocity is a rotational velocity, and the acceleration is a rotational acceleration. 
     In order to provide these definitions, the present invention systematically provides and identification of the joint kinematics for the feet, lower legs, upper legs, pelvis, spine, shoulders and arms. Historically, kinematic models of the human skeleton have been formed as abstractions. In the current invention, the kinematic models are defined using Denavit-Hartenberg conventions, and the dynamic models are defined using the Jacobian conventions. 
     The present invention, in one aspect, provides systems and methods relating to an intelligent analytical system which uses user discrete characteristic data acquired from wearable and other mobile devices. The system has biometric authenticated data integrity, allows for personalized, group or population-based analysis of characteristic data and is not vulnerable to legacy computing systems, power failures, or unauthorized system access. The system, especially the server connected to multiple health or fitness-based databases, has the facility and learning capacity to assess and analyze a user&#39;s unique biometric loop signature data and other characteristics extrapolated from such loop signature data such as step count, step and stride length, stride to stride variability, center of force at any time interval for comparison against stored user data and kinematic chain models describing nominal gait, supination and pronation, and excessive supination and excessive pronation of the foot. The analytical system has the capability for targeted analysis of groupings of populated data characteristics such as gender, nationality, age, height, weight, foot shape such as Germanic or Celtic, foot type such as high arch or flat arch, posture, health and fitness. This analysis may be performed for research in movement and mobility biomarkers. 
     The present invention, in another aspect, provides the specification of a mathematical formalism that accurately specifies the position of each joint, the velocity of each joint, and the acceleration of each joint. When a joint is prismatic, then the velocity is a linear velocity and the acceleration is a linear acceleration. When a joint is rotational, then the velocity is a rotational velocity, and the acceleration is a rotational acceleration. 
     In order to provide these definitions, the invention systematically identifies the joint kinematics for the feet, lower legs, upper legs, pelvis, spine, shoulders and arms. Historically, kinematic models of the human skeleton have been formed as abstractions. In the current invention, the kinematic models are defined using Denavit-Hartenberg conventions, and the dynamic models are defined using the Jacobean conventions. The invention supports a number of skeletal joint types, including:
         Pivot joints enable side-to-side motion. A pivot joint provides for rotation around only one axis. One bone rotates around another within a concave ring formed in the second bone. This ring is lined with a ligament to make the movement smooth. A pivot joint is what enables the neck to rotate to the Left and right and the forearm to make a rotating motion.   Hinge joints enable the bending of limbs. Hinge joints make it possible for limbs to flex and extend along only one axis. The bones fit together perfectly, one convex and the other concave. Elbows, fingers and toes are hinge joints. Certain hinge joints are more complicated to provide limited motion in other directions and are referred to as modified hinge joints. Multiple bones meet at the knee and ankle joints, making them more complex. The resulting structure allows for slight rotation of the knee and circular movement of the ankle.   Ball and Socket Joints enable complex rotation. Ball and socket joints are the most mobile, allowing a wide range of motion. These are the shoulder and hip joints. The bones in these joints fit together with a spherical bone sitting inside another bone that has a concave depression. This structure allows for bending and circular movement as well as rotation of the limb.   Condyloid joints enable joint twisting and bending. Condyloid or ellipsoidal joints are ball and socket joints that are elliptical rather than round, allowing bending and circular movement but rendering rotation impossible. This provides movement in two planes: bending and flexing as a hinge joint as well as a certain amount of rotation. These joints are found in the wrist and the base of the index finger.   Saddle joints are uniquely shaped and provide specifically constrained range of motion. Saddle joints are similar to condyloid joints, but the connecting bones are shaped more like interlocking saddles. This allows for a greater range of motion than hinge joints but does not allow complete rotation like ball and socket joints provide. The thumb is the best example of this.   Gliding Joints support the smooth slipping or sliding of two flat bones against each other and can freely glide past each other in any direction. Gliding joints are found in wrists, ankles and the spine.   Synovial joints enable the human body to move. These complicated connectors make it possible to move from place to place and to eat, work, and play. More than simply places where bones connect, they are a complicated assembly of bone, cartilage and fluid, held together with ligaments and tendons that connect to the muscles that make motion possible.   The spine pays a distinct role in the formation of Gait and is constrained by the inherent range of motion in each of the spinal joints.       

     Referring to  FIG. 1 , a block diagram of one embodiment of the present invention is illustrated. As can be seen, the system  10  includes a sensor module  20  coupled to a data processing module  30  and which may receive data from a storage module  40 . In broad terms, the sensor module  20 , having multiple sensors  20 A, generates biometric data from the sensors (biometric data based on the user&#39;s gait) which is then sent to the data processing module  30 . The data processing module  30  then processes the biometric data and retrieves signature data from the storage module  40 . The signature data comprises data that was previously gathered from the user who is currently being diagnosed. The data processing module then compares the signature data with the biometric data gathered from the multiple sensors. If there are differences between the newly gathered data and the previously gathered data, the data processing module then determines if the differences are in line with known patterns which would indicate progression or regression of known user conditions of disease, sickness or injury. This conclusion would be based on gait-based biometric data which would indicate an improvement or decline in health and fitness or new health conditions. 
     As well, the system includes a communications module  50  that is coupled to the data processing module  30 . The communications module  50  sends and/or receives communications regarding the comparison between the signature data and the data gathered from the sensors. The communications module  50  sends the data gathered to the data processing module  30  such that further data processing is performed remote from the user and/or the sensor module  20 . The further data processing may be performed by a personal mobile device or by a server remote from the user&#39;s location and coupled to a network of distributed databases. By off-loading most of the post biometric authentication processing to a personal mobile device or to the remote server, the insole system does not need much processing capability. In one implementation, the sensor module simply operates as a data gathering device and all data processing is performed remote from the sensor module and/or the user. The data gathered by the sensor module may be transmitted to the data processing module for processing and/or to a database for later processing or to be part of one or more data sets. 
     It should be noted that the sensor module  20  has multiple sensors which gather data regarding a person&#39;s gait as well as other discrete user characteristics such as weight, step count, stride length, stride to stride variability, centre of force, applied force, cadence and lateral distance between the feet. In one embodiment, the sensor module is an insole positioned inside the user&#39;s shoe, with the insole having multiple discrete force sensors that detect the amount of force or pressure exerted on a section or region of the insole. With multiple regions on the insole and at least one sensor positioned on each region, a user&#39;s gait can be profiled as being the amount of pressure that the user exerts on each region over time as the user takes a step and correlated with embedded accelerometer and gyroscope data derived from the chip based computing module housed within the insole apparatus. A variant of this sensor module would have at least one strain gauge positioned such that the pressure exerted on each of the multiple regions of the foot are detected by the gauge with each region corresponding to a section of the strain gauge. With such an arrangement, each section of the strain gauge thus acts as a different discrete sensor and correlated with embedded accelerometer and gyroscope data derived from the chip-based computing module housed within the insole apparatus. 
     It should be noted that, in one embodiment, two insoles are used per user. This way, gait data may be gathered for each user foot. Data gathered from the user&#39;s left foot may be processed differently from data gathered from the user&#39;s right foot. The data gathered from each foot may then be combined to determine characteristics such as, for example, step count, cadence, velocity, centre of force, load bearing weight, standing weight and stride length. Alternatively, another embodiment only uses a single insole such that only one set of data is gathered per user. While the description below relates to a single insole, for a two-insole embodiment, both insoles would be similar to one another and would, preferably, each conform to the description and principles outlined below. 
     Referring to  FIGS. 2A and 2B , a schematic illustration of a number of discrete pressure zones on an insole is illustrated.  FIG. 2A  shows an imprint of a human foot and the unique pressure points for a specific person.  FIG. 2B  illustrates the location of 8 specific pressure zones or areas on one embodiment of a pressure sensing insole. Each zone in  FIG. 2B  has a pressure sensing pad or a number of sensors assigned to it such that the pressure exerted on each zone can be measured. A variant of this sensor module would have, instead of discrete sensor pads at each zone, a single strain gauge positioned as described above. 
     In the above embodiment, each sensor in the sensor module produces a signal linearly proportional to the force being applied to the sensor. Preferably, each sensor or zone would have a data channel dedicated to its readings for transmitting those readings to the data processing module. Alternatively, in one implementation, the readings can be time division multiplexed on to a single data line from the sensor module to the data processing module. In this implementation, the data is passed through a single A/D converter to produce multiplexed channels, one for each sensor. Of course, while there are eight zones in  FIG. 2B , other variants may have more than or less than eight zones. 
     Regarding the data stream produced when the user is walking, in one embodiment, each sensor produces several hundred samples equating to approximately ten steps taken by the user. This data stream is then saved and examined by the data processing module and the actual step points are determined. Each step is identified, and the saved data stream resampled at a precise rate of approximately 100 samples per step. 
     It should be noted that multiple parameters regarding the user&#39;s gait can be extracted from the data produced by the sensor module depending on the type of sensors used in the sensor module. These parameters can then be used as points of comparison with the signature (or characteristic) data mentioned above. Some of these parameters may be:
         Actual forces   Relative (normalized) forces.   Ratios between the peak forces in the eight sensor zones   Relative timing between forces on each sensor (strike and release sequence)   Average rate of change of force on each sensor zone   Maximum rate of change of force on each sensor zone   Frequency spectrum of the waveform from each sensor (ratio of values of harmonics derived from a Fourier transform)   Heel strike and toe lift off impact forces in the three axes.   Data waveform shape matching (waveform shape matching)       

     The parameters extracted from the data stream may then be compared directly or indirectly with the signature data noted above. 
     In one comparison scheme, the parameters extracted are used to derive a shape or loop, the characteristics of which can the compared with characteristics of a signature loop or shape. The use of a loop or shape allows for an indirect comparison between the data read by the sensor module and the signature or characteristic data. As well, it allows for more complex comparison schemes and for easier use of tolerances in the comparison. 
     For this comparison scheme, data from two different sensors are read by the data processing module. The two data sets (one from a first sensor and a second from a second sensor) are correlated with one another to synchronize the readings. This is done so that the data readings are synchronized in their time indices. Once synchronized, readings taken at approximately the same time index are matched with one another. Thus the result is that a data reading from sensor A taken at time t1 is mated with a data reading from sensor B taken at time t1. The mating step results in a set of pairs of data readings from two different sensors. 
     It should be noted that a preferable preliminary step to the correlation step is that of applying a low pass filter to both sets of data. Such a low pass filter would remove the low frequency components of the signals and would provide cleaner and easier to process signals. 
     As an example of the processing performed on the data streams received from the sensor modules,  FIGS. 3-8  are provided to aid in the understanding of the process. Prior to any processing, data streams are first received from all of the sensors for a given fixed duration. For each sensor, the data stream for the given duration is saved by the data processing module. The resulting waveform for each sensor is then partitioned to determine discrete steps taken by the user. If the sensors are force/pressure sensors, this partitioning may be done by searching for peaks and valleys in the waveform. Each peak would denote a maximum force applied to the sensor and each valley would denote a minimum (if not absence) of force. Each step can then be seen as two valleys with a peak in between, representing the user&#39;s foot in the air, the actual step, and then user lifting his/her foot again. Alternatively, depending on how the system is configured, each step might be seen as two peaks bookending a valley. 
     Referring to  FIG. 3 , two raw data streams is shown at the bottom of the plot. After a low pass filter is applied to the signals, the smoother waveforms are shown at the top half of  FIG. 3 . From  FIG. 3 , one can see maximum force applied to the force pads for the two steps captured by the waveforms. 
     Once the discrete steps have been delineated in the data received from each of the sensors, each step for each sensor is then resampled to arrive at a predetermined number of data samples for each step. For the resampling, each sample is for a predetermined time frame and at a predetermined point in time in the current step. As an example, if each step lasts approximately 0.1 sec and 100 samples per step are desired, then the first sample is taken at the first one thousandth of a second in the waveform and the second sample is taken at the second one thousandth of a second and so on and so forth. This method essentially synchronizes all the samples such that it would be simple to determine all samples (from all the sensor readings) taken at the first one thousandth of a second or all samples taken at the first fiftieth one thousandths of a second as the relevant samples would all be similarly time indexed. 
     Once the different data waveforms from the different sensors have been synchronized, any two of the sensors and the data they produced can be selected for comparison with the signature data noted above and which may be stored in the data storage module. Depending on the configuration of the system, the signature or characteristic data stored in the data storage module may take numerous forms. In one example, multiple data sets/pairs (either filtered or as raw data) from the user may be stored so that a signature loop may be derived from the signature data whenever the characteristics of that signature loop are required. For this example, all the data pairs from all sensors would be stored so that any two sensors may be selected. Alternatively, the specific characteristics of the signature loop may be stored as the signature or characteristic data if one wanted to dispense with determining the signature loop every time a comparison needs to be made. As another alternative, only the data relating to the average signature loop derived from the user may be stored as signature data. Of course, if multiple sensors are to be used, then most possible average signature loops from the user data would be stored. 
     In one other alternative, all the raw data (either filtered or not) from the user&#39;s steps may be stored as signature data. Such a configuration would allow for the greatest amount of flexibility as the system could randomly select any two of the sensors to be used and the signature data from the user would be available for those two sensors. As noted above, this configuration would require that the signature loop be calculated every time a comparison is required. The signature data may, if desired, be stored in encrypted format. 
     Once two of the sensors are selected from the sensors available in the sensor module (in this example the sensor module has 8 sensors, one for each of the eight zones illustrated in  FIG. 2B ), the resampled data for those two sensors are then mated with one another. This means that each time indexed sampling will have two points of data, one for the first sensor and another for the other sensor. These pairs of sensor readings can thus be used to create a characteristic loop. As an example, if sensors A and B are used and n denotes an index, then A[n] denotes the nth sampled reading from the waveform received from sensor A for a specific step. Similarly, B [n] is the nth sampled reading from the waveform received from sensor B for the same specific step. {A[n], B[n]} thus constitutes a data pair for the nth reading for that particular step. Plotting all the data pairs for a particular step, with readings from one sensor being on one axis and readings from the other sensor on the axis, results in an angled loop-like plot (see  FIGS. 4-8  as examples). For pressure/force readings, this is not surprising as the force exerted by the foot in a particular step increases to a maximum and then decreases to as minimum as the person increases the weight on the place on the foot and then removes that weight as the step progresses. 
     Once the data pairs have been created, a plot of the resulting loop can be made. As noted above,  FIG. 3  shows the waveforms for two signals—the lower waveform being the raw data stream waveforms for 2 signals and the upper waveforms for the same 2 signals after a low pass filter has been applied.  FIG. 4  shows a plot of the two sets of waveforms in  FIG. 3 . One loop in  FIG. 4  is derived from the raw signal waveforms in  FIG. 3  while the other loop is derived from the low pass filtered waveform in  FIG. 3 . As can be seen in  FIG. 4 , a smoother loop is produced by the low-pass filtered signals. 
     It should be noted that the x-axis in  FIG. 4  contains the values gathered from the first sensor selected while the y-axis contains the values gathered from the second selected sensor. It should be noted that while the embodiment discussed uses only a pair of sensors, the concept is applicable for 3, 4, or any number of sensors. If data from 3 sensors were used, then, instead of a 2D loop, a 3D loop may be created as a characteristic loop. 
     It should be noted that a loop can be formed for each one of the steps captured by the sensors. An averaged loop can be derived from the various loops formed from all the steps captured by the sensors. Referring to  FIG. 5 , the various loops from the various steps can be seen on the plot. An average loop (see darker loop in  FIG. 5 ) is derived from all the loops captured using the low pass filtered waveforms. Multiple methods may be used to determine the average loop. However, in one embodiment, the points for the average loop are derived by averaging the various readings for each particular time index. As such, if the data pairs are as (An[i],Bn[i]) with An[i] denoting the nth reading for sensor A at time index i and Bn[i] denoting the nth reading for sensor B at time index i, then to derive the data reading for sensor A for the average loop for time index i, one merely averages all the An[i] where n=1, 2, 3, etc., etc. Similarly, for data reading for sensor B for the average loop for time index i, one merely averages all the Bn[i] where n=1, 2, 3, etc., etc. By doing this for all the multiple time indices, an average loop is derived from all the characteristic loops. 
     Once the average loop has been derived, the characteristics of that average loop can be determined. Referring to  FIG. 6 , some of the characteristics of the average loop can be seen. The length of the loop (measured from the origin), the width of the widest part of the loop, and the area occupied by the loop are just some of the characteristics which may be determined from the loop. As well, the direction of the loop (whether it develops in a clockwise or anti-clockwise manner) may also be seen as a characteristic of the loop. Another possible characteristic of the loop may be the angle between a ray from the origin to the farthest point of the loop and one of the axes of the plot. Additional characteristics of these loops may, of course, be used depending on the configuration of the system. 
     As another example of possible loops,  FIG. 7  shows loops resulting from highly correlated data from the sensors. Such highly correlated data may produce loops that, at first glance, may not be overly useful. However, even such lopsided loops may yield useful characteristics. As an example, the amplitude from the furthest point may be used for an initial assessment of static of dynamic weight distribution. 
     Once the average loop for the steps captured by the sensors is determined, the characteristics for this average loop can be derived. Once derived, the same process is applied to the signature data stored in the storage module. The characteristics for the resulting signature loop (from the signature data) are then compared to the characteristics of the average loop from the data acquired from the sensors. 
     Referring to  FIG. 8 , a comparison of two average loops from the gait of two individuals is illustrated. As can be seen, the characteristics of the two loops are quite different. One loop is clearly larger (more area), longer (length of loop), and wider (width at widest of the loops) than the other loop. It should be noted that custom tolerances can be applied to the comparison. Depending on the tolerance applied, the comparison can be successful (the characteristics match within the tolerances) or unsuccessful (even within the tolerances, there is no match). It should be noted that, as noted above, comparisons can be made for loops from a single user with data taken at different times. As an example, a user may have gait data taken at a first use of the system. Later gait data sets can then be taken for the same user at subsequent uses of the system. The loops derived from the initial gait data set and the subsequent gait data sets can then be compared to determine how a particular fitness program, treatment regimen, or physical condition has positively or negatively affected that user&#39;s gait over time. 
     Regarding tolerances, these can be preprogrammed into the system and can be determined when the signature data is gathered. As an example, a tolerance of 15% may be acceptable for some users while a tolerance of only 5% may be acceptable. This means that if the calculated characteristic of the average loop is within 15% of the calculated characteristic of the signature loop, then a match is declared. A match would indicate that there is no relevant difference between the loops being compared. Similarly, if a tolerance of only 5% is used, then if the calculated characteristic of the average loop is within 5% of the calculated characteristic of the signature loop, then a match is declared. Of course, if the calculated characteristic of the average loop is not within the preprogrammed tolerance of the calculated characteristic of the signature loop, then a non-match is declared. A non-match would indicate that there is a relevant difference between the loops being compared. A match may indicate that, for example, a fitness program, treatment regimen, or condition has not affected a user&#39;s gait between the time the first set of gait data was gathered to the time the second set of gait data was gathered. A non-match may, of course, indicate that the fitness program, treatment regimen, or condition has affected the user&#39;s gait. 
     It should also be noted that, in addition to the tolerances noted above, the system may use a graduated system of matches or matching. This would mean that a level of confidence may be assigned to each match, a high level of confidence being an indication that there is a higher likelihood that there is a match between the two sets of data derived from the average loop and the signature loop. A match can then be declared once the level of confidence assigned is higher than a predetermined level. A non-match can similarly be declared once the level of confidence is lower than a predetermined level. A level of indecision can be declared when the level of confidence is between the two pre-set levels for match and non-match. If a set of data falls within the gray area or an area of indecision between the two pre-set levels, then more data can be retrieved from the sensors and this data can be processed as above to arrive at a determination of a match or a non-match. 
     It should further be noted that, as an alternative, instead of matching or not matching two loops derived from a user&#39;s gait data, the amount of difference between the two loops can be determined. A significant difference between the characteristics of the two loops, preferably derived from data gathered from the same user using the same sensors in the sensor module at different times, would indicate a change of some sort. A significant difference between such two loops would indicate a significant change from the time the first data set was gathered to the time the second data set was gathered. As noted above, this could indicate that a fitness program, treatment regimen, or condition was having an effect on the user&#39;s gait. It may also indicate that a user&#39;s physical or medical condition is either progressing or regressing. The characteristics for which a difference may be found may, as noted above, include the size of the loops, the angle of the loops to one of the axes of the plot, the perimeter of the loops, the area covered by the loops, as well as other characteristics. A tolerance may, of course, be built into the comparison subroutine. As an example, if the tolerance is set at 2%, if a characteristic of two loops are within 1% (i.e. less than 2%) of each other (e.g. the sizes of the two loops) then no difference is concluded. 
     For greater clarity, the difference between two loops may be quantified and, depending on how great the differences are, alarms or other steps may be taken. As an example, if the area of a loop derived from a user&#39;s initial data set is compared with the area of a loop derived from a data set gathered a few months later, the differences may be significant. If there is no appreciable difference, then one can conclude that no change has occurred in the user&#39;s condition. If, on the other hand, the second data set has a much larger area (e.g. 25% greater area than the area covered by the loop from the first data set), this may indicate that the user is walking slower or that the user is placing more pressure on his feet with each step. Depending on the user&#39;s physical condition, this may indicate a progression (getting worse) or a regression (getting better) of that condition. It may also indicate that a fitness program or treatment regimen being used may or may not be effective. A threshold may thus be programmed so that if the difference in value of a characteristic being compared between two loops exceeds a specific amount or percentage, an alarm may be activated. 
     Refined comparison of gait data to existing gait models can also be performed. Kinematic and dynamic models are developed by specifying kinematic models of the joints contributing to ambulation, including the phalanges, metatarsals, tarsals, tibia/fibula, femur, pelvis, spinal vertebrae, and arm motion. Loop data is translated into pressure indicators, where pressure is propagated through the kinematic model. 
     The kinematic data is expressed as joint position and joint angles for every bone that participates in ambulation. 
     Dynamic joint data is modeled using a Jacobian matrix of partial differential equations of the joint velocities and joint accelerations for each joint that participates in ambulation. With the kinematic and dynamic models, comparisons of current data, registration data, and nominal data can be conducted to confirm that a user&#39;s performance has not altered from a known state. 
     In addition, kinematic and dynamic joint data comparisons can be made to known gait models, such as those derived from particular diseases and their unique gait data observed and collected from known patient observations. Models derived from these observations are compared by position in X, Y, Z, linear velocity in X′, Y′, Z′, linear acceleration in X″, Y″, Z″, where prismatic or sliding joints are observed. In the case of rotational joints, the joint positions in θ x , θ y , and θ z , joint velocities, in ωx, ωy, and ωz, and joint accelerations in α x , α y , and α z  are used. 
     In addition, the coordinate reference frame for each of the 187 joints in the analytical system configuration is expressed below. This analytical system configuration represents shuffling, walking, running, jumping, or dancing behaviour. As well, the configuration may be used to represent biomarkers for disease detection, biomarkers for disease progression or regression, injury, sickness and/or side effects of prescribed pharmacologies. The description of each of the 187 joints and the Jacobian matrices for each joint as provided below. 
     Joint 1 connects the Right Foot, 1 st  Toe, Hallux, Distal Phalange to the Right Foot, 1 st  Toe, Proximal Phalange. The d 1  link length of the Hallux runs collinear with the hallux centroid running from the Hallux base to the tip of the 1 st  Proximal Phalange along the X axis. The matrices for this joint are: 
     
       
         
           
             
               A 
               1 
             
             = 
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             
                               d 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               a 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               A 
               1 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                        
                       cos 
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       
                         a 
                         1 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       
                         a 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         1 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         1 
                       
                     
                   
                   
                     
                       d 
                       1 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 2 connects the Right foot, 2 nd  toe, Distal Phalange to the Right foot 2 nd  toe Middle Phalange. The d 2  link length of the 2 nd  Distal Phalange runs collinear with the 2 nd  Distal Phalange centroid running from the 2 nd  Distal Phalange base to the tip of the 2 nd  Middle Phalange along the X axis. The matrices for this joint are: 
     
       
         
           
             
               A 
               2 
             
             = 
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 2 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 2 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             
                               d 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               a 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               A 
               2 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                        
                       cos 
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       
                         a 
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       
                         a 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         2 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         2 
                       
                     
                   
                   
                     
                       d 
                       2 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 3 connects the Right foot, 3 rd  toe, Distal Phalange to the Right foot, 3 rd  toe Middle Phalange. The d 3  link length of the 3 rd  Distal Phalange runs collinear with the 3 rd  Distal Phalange centroid running from the 3 rd  Distal Phalange base to the tip of the 3 rd  Middle Phalange along the X axis. The matrices for this joint are: 
     
       
         
           
             
               A 
               3 
             
             = 
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 3 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 3 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             
                               d 
                               3 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               a 
                               3 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               A 
               3 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                        
                       cos 
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       
                         a 
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       
                         a 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         3 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         3 
                       
                     
                   
                   
                     
                       d 
                       3 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 4 connects the Right foot, 4 th  toe, Distal Phalange to the Right foot, 4 th  toe, Middle Phalange. The d 4  link length of the 4 th  Distal Phalange runs collinear with the 4 th  Distal Phalange centroid running from the 4 th  Distal Phalange base to the tip of the 4 th  Middle Phalange along the X axis. 
     
       
         
           
             
               A 
               4 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               4 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               4 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               4 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               4 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             4 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           4 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       4 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               
                                 a 
                                 4 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               
                                 a 
                                 4 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 4 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 4 
                               
                             
                           
                           
                             
                               d 
                               4 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 5 connects the Right foot, 5 th  toe, Distal Phalange to the Right foot, 5 th  toe, Middle Phalange. The d 5  link length of the 5 th  Distal Phalange runs collinear with the 5 th  Distal Phalange centroid running from the 5 th  Distal Phalange base to the tip of the 5 th  Middle Phalange along the X axis. The matrices for this joint are as follows: 
     
       
         
           
             
               A 
               5 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               5 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               5 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               5 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               5 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             5 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           5 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       5 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               
                                 a 
                                 5 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               
                                 a 
                                 5 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 5 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 5 
                               
                             
                           
                           
                             
                               d 
                               5 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 6 connects the Right foot, 2 nd  toe, Middle Phalange to the Right foot, 2 nd  toe, Proximal Phalange. The d 7  link length of the 2 nd  Middle Phalange runs collinear with the 2 nd  Middle Phalange centroid running from the 2 nd  Middle Phalange base to the tip of the 2 nd  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               6 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               6 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               6 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               6 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               6 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             6 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           6 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       6 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               
                                 a 
                                 6 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               
                                 a 
                                 6 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 6 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 6 
                               
                             
                           
                           
                             
                               d 
                               6 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 7 connects the right foot, 3 rd  toe, Middle Phalange to Right foot, 3 rd  toe, Proximal Phalange. The d 7  link length of the 3 rd  Middle Phalange runs collinear with the 3 rd  Middle Phalange centroid running from the 3 rd  Middle Phalange base to the tip of the 3 rd  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               7 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               7 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               7 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               7 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               7 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             7 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           1 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       7 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               
                                 a 
                                 7 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               
                                 a 
                                 7 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 7 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 7 
                               
                             
                           
                           
                             
                               d 
                               7 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 8 connects the Right foot, 4 th  toe, Middle Phalange to the Right foot, 4 th  toe, Proximal Phalange. The d 8  link length of the 4 th  Middle Phalange runs collinear with the 4 th  Proximal Phalange centroid running from the 4 th  Middle Phalange base to the tip of the 4 th  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               8 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               8 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               8 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               8 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               8 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             8 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           8 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       8 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               
                                 a 
                                 8 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               
                                 a 
                                 8 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 8 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 8 
                               
                             
                           
                           
                             
                               d 
                               8 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 9 connects the Right foot, 5 th  toe, Middle Phalange to the Right foot, 5 th  toe Proximal Phalange. The d 9  link length of the 5 th  Middle Phalange runs collinear with the 5 th  Proximal Phalange centroid running from the 5 th  Middle Phalange base to the tip of the 5 th  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               9 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               9 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               9 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               9 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               9 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             9 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           9 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       9 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               
                                 a 
                                 9 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               
                                 a 
                                 9 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 9 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 9 
                               
                             
                           
                           
                             
                               d 
                               9 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 10 connects the Right foot 1 st  toe Proximal Phalange to the Right foot, 1 st  toe Metatarsal. The d 10  link length of the 1 st  Proximal Phalange runs collinear with the 1 st  Proximal Phalange centroid running from the 1 st  Proximal Phalange base to the tip of the 1 st  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               10 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               10 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               10 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               10 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               10 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             10 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           10 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       10 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               
                                 a 
                                 10 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               
                                 a 
                                 10 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 10 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 10 
                               
                             
                           
                           
                             
                               d 
                               10 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 11 connects the Right foot, 2 nd  toe, Proximal Phalange to the Right foot, 2 nd  toe Metatarsal. The d 10  link length of the 2 nd  Proximal Phalange runs collinear with the 2 nd  Proximal Phalange centroid running from the 2 nd  Proximal Phalange base to the tip of the 2 nd  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               11 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               11 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               11 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               11 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               11 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             11 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           11 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       11 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               
                                 a 
                                 11 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               
                                 a 
                                 11 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 11 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 11 
                               
                             
                           
                           
                             
                               d 
                               11 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 12 connects the Right foot, 3 rd  toe, Proximal Phalange to the Right foot, 3 rd  toe Metatarsal. The d 12  link length of the 3 rd  Proximal Phalange runs collinear with the 3 rd  Proximal Phalange centroid running from the 3 rd  Proximal Phalange base to the tip of the 3 rd  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               12 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               12 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               12 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               12 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               12 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             12 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           12 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       12 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               
                                 a 
                                 12 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               
                                 a 
                                 12 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 12 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 12 
                               
                             
                           
                           
                             
                               d 
                               12 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 13 connects the Right foot, 4 th  toe Proximal Phalange to the Right foot, 4 th  toe Metatarsal. The d 13  link length of the 4 th  Proximal Phalange runs collinear with the 4th Proximal Phalange centroid running from the 4 th  Proximal Phalange base to the tip of the 4 th  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               13 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               13 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               13 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               13 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               13 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             13 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           13 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       13 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               
                                 a 
                                 13 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               
                                 a 
                                 13 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 13 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 13 
                               
                             
                           
                           
                             
                               d 
                               13 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 14 connects the Right foot, 5 th  toe Proximal Phalange to the Right foot, 5 th  toe Metatarsal. The d 14  link length of the 5 th  Proximal Phalange runs collinear with the 5 th  Proximal Phalange centroid running from the 5 th  Proximal Phalange base to the tip of the 5 th  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               14 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               14 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               14 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               14 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               14 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             14 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           14 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       14 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               
                                 a 
                                 14 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               
                                 a 
                                 14 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 14 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 14 
                               
                             
                           
                           
                             
                               d 
                               14 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 15 connects the Right foot, 1 st  toe Metatarsal to the Right foot, 2 nd  toe Metatarsal. The d 15  link length of the Right big toe Metatarsal runs collinear with the 1 st  Metatarsal centroid running from the 1 st  Metatarsal to the 2 nd  Metatarsal along the Y axis. 
     
       
         
           
             
               A 
               15 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               15 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               15 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               15 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               15 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             15 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           15 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       15 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               
                                 a 
                                 15 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               
                                 a 
                                 15 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 15 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 15 
                               
                             
                           
                           
                             
                               d 
                               15 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 16 connects the Right foot, 2 nd  toe Metatarsal to the Right foot 3 rd  toe Metatarsal. The d 16  link length of the Right 2 nd  Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal to the 3 rd  Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               16 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               16 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               16 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               16 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               16 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             16 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           16 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       16 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               
                                 a 
                                 16 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               
                                 a 
                                 16 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 16 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 16 
                               
                             
                           
                           
                             
                               d 
                               16 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 17 connects the Right foot, 3 rd  toe Metatarsal to the Right foot 4 th  toe Metatarsal. The d 17  link length of the Right 4 th  Metatarsal runs collinear with the 3 rd  Metatarsal centroid running from the 3 rd  Metatarsal to the 3 rd  Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               17 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               17 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               17 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               17 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               17 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             17 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           17 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       17 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               
                                 a 
                                 17 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               
                                 a 
                                 17 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 17 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 17 
                               
                             
                           
                           
                             
                               d 
                               17 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 18 connects the Right foot, 4th toe Metatarsal to the Right foot, 5 th  toe Metatarsal. The d 18  link length of the Right 4 th  Metatarsal runs collinear with the 4th Metatarsal centroid running from the 4 th  Metatarsal to the 4th Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               18 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               18 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               18 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               18 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               18 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             18 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           18 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       18 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               
                                 a 
                                 18 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               
                                 a 
                                 18 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 18 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 18 
                               
                             
                           
                           
                             
                               d 
                               18 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 19 connects the Right foot, 1 st  toe Metatarsal to the Right foot medial cuneiform. The d 19  link length of the Right big toe Metatarsal runs collinear with the 1 st  Proximal Phalange centroid running from the 1 st  Proximal Phalange base to the tip of the 1 st  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               19 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             19 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             19 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             19 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             19 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       
                         d 
                         19 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               a 
                               19 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               19 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               α 
                               19 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               19 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               19 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
 
                     
                      
                     
                       
                         A 
                         19 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   19 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   19 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   19 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   19 
                                 
                               
                             
                             
                               
                                 d 
                                 19 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     Joint 20 connects the Right foot, 2 nd  toe Metatarsal to Right foot, Medial Cuneiform. The d 20  link length of the right 2 nd  toe metatarsal runs collinear with the 2 nd  metatarsal centroid running from the 2 nd  Metatarsal base to the tip of the Right Medial Cuneiform along the X axis. 
     
       
         
           
             
               A 
               20 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             20 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             20 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             20 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             20 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           20 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 20 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 20 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 20 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 20 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       20 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             
                               a 
                               20 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             
                               a 
                               20 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               20 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               20 
                             
                           
                         
                         
                           
                             d 
                             20 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 21 connects the Right foot, 2 nd  toe Metatarsal to the Right foot Intermediate cuneiform. The d 21  link length of the Right 2 nd  toe Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal base to the tip of the Right Intermediate Cuneiform along the X axis 
     
       
         
           
             
               A 
               21 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             21 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             21 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             21 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             21 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           21 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 21 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 21 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 21 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 21 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 21 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       21 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             
                               a 
                               21 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               21 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               21 
                             
                           
                         
                         
                           
                             d 
                             21 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 22 connects the Right foot, 2 nd  toe Metatarsal to the Right foot Lateral Cuneiform. The d 22  link length of the Right 2 nd  toe Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal base to the tip of the Right Lateral Cuneiform along the X axis 
     
       
         
           
             
               A 
               22 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             22 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             22 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             22 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             22 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           22 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 22 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 22 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 22 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 22 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 22 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       22 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             
                               a 
                               22 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             
                               a 
                               22 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               22 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               22 
                             
                           
                         
                         
                           
                             d 
                             22 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 23 connects the Right foot, 3 rd  toe Metatarsal to the Right fool Lateral Cuneiform. The d 23  link length of the Right 3 rd  Metatarsal runs collinear with the 3rd Metatarsal centroid running from the 3 rd  Metatarsal base to the tip of the Right Lateral Cuneiform along the X axis 
     
       
         
           
             
               A 
               23 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             23 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             23 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             23 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             23 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           23 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 23 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 23 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 23 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 23 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 23 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       23 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             
                               a 
                               23 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             
                               a 
                               23 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               23 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               23 
                             
                           
                         
                         
                           
                             d 
                             23 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 24 connects the right foot, 4 th  toe metatarsal to the Right foot Lateral Cuneiform. The d 24  link length of the Right 4 th  Metatarsal runs collinear with the 4 th  Metatarsal centroid running from the 4 th  Metatarsal base to the tip of the Right Lateral Cuneiform along the X axis. 
     
       
         
           
             
               A 
               24 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             24 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             24 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             24 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             24 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           24 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             a 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             24 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             α 
                             24 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             α 
                             24 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             24 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Joint 25 connects the Right foot, 4th toe Metatarsal to the Right foot cuboid bone. The d 25  link length of the Right 4 th  Metatarsal runs collinear with the 4th Metatarsal centroid running from the 4 th  Metatarsal base to the tip of the Right Cuboid along the X axis. 
     
       
         
           
             
               A 
               25 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             25 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             25 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             25 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             25 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           25 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 25 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 25 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 25 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 25 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 25 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       25 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             
                               a 
                               25 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             
                               a 
                               25 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               25 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               25 
                             
                           
                         
                         
                           
                             d 
                             25 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 26 connects the Right foot, 5th toe Metatarsal to the Right foot cuboid bone. The d 26  link length of the Right 5th Metatarsal runs collinear with the 5th Metatarsal centroid running from the 5 th  Metatarsal base to the tip of the Right Cuboid along the X axis. 
     
       
         
           
             
               A 
               26 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             26 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             26 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             26 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             26 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           26 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 26 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 26 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 26 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 26 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 26 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       26 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             
                               a 
                               26 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             
                               a 
                               26 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               26 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               26 
                             
                           
                         
                         
                           
                             d 
                             26 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 27 connects the Right foot, Medial Cuneiform to the Right foot Intermediate Cuneiform. The d 27  link length of the Right Medial Cuneiform runs collinear with the Right Medial Cuneiform running from the Right Medial Cuneiform base to the tip of the Right Intermediate Cuneiform along the X axis. 
     
       
         
           
             
               A 
               27 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             27 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             27 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             27 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             27 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           27 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 27 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 27 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 27 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 27 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       27 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             
                               a 
                               27 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             
                               a 
                               27 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               27 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               27 
                             
                           
                         
                         
                           
                             d 
                             27 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 28 connects the Right foot, Intermediate Cuneiform to the Right foot Lateral Cuneiform. The d 28  link length of the Right Intermediate Cuneiform runs collinear with Right Intermediate Cuneiform running from the Intermediate Cuneiform base to the tip of the Lateral Cuneiform along the X axis. 
     
       
         
           
             
               A 
               28 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             28 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             28 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             28 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             28 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           28 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 28 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 28 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 28 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 28 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       28 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             
                               a 
                               28 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             
                               a 
                               28 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               28 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               28 
                             
                           
                         
                         
                           
                             d 
                             28 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 29 connects the Right foot, Lateral Cuneiform to the Right foot Cuboid. The d 29  link length of the Right Lateral Cuneiform runs collinear with the Right Lateral Cuneiform centroid running from the Lateral Cuneiform base to the tip of the Cuboid along the Y axis. 
     
       
         
           
             
               A 
               29 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             29 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             29 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             29 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             29 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           29 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 29 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 29 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 29 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 29 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       29 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             
                               a 
                               29 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             
                               a 
                               29 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               29 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               29 
                             
                           
                         
                         
                           
                             d 
                             29 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 30 connects the Right foot, Medial Cuneiform bone to the Right foot Navicular bone. The d 30  link length of the Right Medial Cuneiform runs collinear with the Right Medial Cuneiform running from the Right Medial Cuneiform base to the tip of the Right Navicular along the X axis. 
     
       
         
           
             
               A 
               30 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             30 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             30 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             30 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             30 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           30 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 30 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 30 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 30 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 30 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       30 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             
                               a 
                               30 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             
                               a 
                               30 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               30 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               30 
                             
                           
                         
                         
                           
                             d 
                             30 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 31 connects the Right foot, Intermediate Cuneiform bone to the Right foot navicular bone. This joint is both rotational of the Right Intermediate Cuneiform bone about the Right navicular. The d 31  link length of the Right Intermediate Cuneiform runs collinear with the Right medial Cuneiform running from the Intermediate Cuneiform to the tip of Navicular along the X axis. 
     
       
         
           
             
               A 
               31 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             31 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             31 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             31 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             31 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           31 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 31 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 31 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 31 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 31 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 31 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       31 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             
                               a 
                               31 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               31 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               31 
                             
                           
                         
                         
                           
                             d 
                             31 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 32 connects the Right foot, Lateral Cuneiform bone to the Right foot Navicular bone. This joint is both rotational of the Right Lateral Cuneiform bone about the Right Navicular in the X plane, and rotation of the Right Lateral Cuneiform bone about the Right Navicular in the Y plane, in an oblong interface supporting rotational motion in two direction. There is no linear motion in either the X plane or the Y plane in this joint. 
     
       
         
           
             
               A 
               32 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             32 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             32 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             32 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             32 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           32 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 32 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 32 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 32 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 32 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       32 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             
                               a 
                               32 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             
                               a 
                               32 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               32 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               32 
                             
                           
                         
                         
                           
                             d 
                             32 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 33 connects the right foot, Navicular Cuneiform to the Right foot Cuboid. The d 33  link length of the right Navicular runs collinear with the Navicular centroid running from the Navicular to the Cuboid along the Y axis. 
     
       
         
           
             
               A 
               33 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             33 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             33 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             33 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             33 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           33 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 a 
                                 1 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 33 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 33 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 33 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 33 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       33 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             
                               a 
                               33 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             
                               a 
                               33 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               33 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               33 
                             
                           
                         
                         
                           
                             d 
                             33 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 34 connects the Right foot, Navicular bone to the Right foot Talus bone. This joint is a rotational joint with respect to the Navicular bone and the Talus bone in the Z plane, and it is also a rotational joint with respect to the Navicular bone and the Talus bone in the Y plane. There is no linear lateral motion in this joint, rather, the curved structure of the Talus-Navicular interface is curved, thus supporting only rotation. 
     
       
         
           
             
               A 
               
                 3 
                  
                 4 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 3 
                                  
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 3 
                                  
                                 4 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 3 
                                  
                                 4 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 3 
                                  
                                 4 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             
                               3 
                                
                               4 
                             
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             3 
                              
                             4 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 3 
                  
                 4 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           3 
                            
                           4 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           3 
                            
                           4 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           3 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       d 
                       
                         3 
                          
                         4 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 35 connects the right foot, Cuboid to the Right foot Calcaneus. The d 35  link length of the Right Calcaneus runs collinear with the Right Cuboid running from the Cuboid base to the tip of the Calcaneus along the X axis. 
     
       
         
           
             
               A 
               35 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               35 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               35 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               35 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               35 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             35 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           1 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               35 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       
                         a 
                         35 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       
                         a 
                         35 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         35 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         35 
                       
                     
                   
                   
                     
                       d 
                       35 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 36 connects the right foot, Talus to the right foot Calcaneus. The d 36  link length of the right Talus runs collinear with the right Talus running from the Talus to the Calcaneus along the X axis. 
     
       
         
           
             
               A 
               36 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               36 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               36 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               36 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               36 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             36 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           1 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               36 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       
                         a 
                         36 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       
                         a 
                         36 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         36 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         36 
                       
                     
                   
                   
                     
                       d 
                       36 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 37 connects the Right fibula to the Right Talus bone, this is commonly referred to as the ankle joint. The d 37  link length of the Right Fibula runs collinear with the right Fibula running from the Fibula base to the Talus along the X axis. 
     
       
         
           
             
               A 
               37 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               37 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               37 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               37 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               37 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             37 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           37 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               37 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         a 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         a 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       d 
                       37 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 38 connects the Right tibia to the Right Talus bone. The d 38  link length of the Right Tibia runs collinear with the Right Tibia running from the Right Tibia base to the tip of the Right Talus along the X axis. 
     
       
         
           
             
               A 
               38 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               38 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               38 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               38 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               38 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             38 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           38 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               38 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       
                         a 
                         38 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       
                         a 
                         38 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         38 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         38 
                       
                     
                   
                   
                     
                       d 
                       38 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 39 connects the Dorsal end of the Right Tibia to the dorsal end of the Right Fibula to form the Right Dorsal Tibiofibular joint. The d 39  link length of the Right Dorsal Tibia runs collinear with the Right Dorsal Tibia running from the Dorsal Tibia to the Right Dorsal Fibula along the Y axis. 
     
       
         
           
             
               A 
               39 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               39 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               39 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               39 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               39 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             39 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           39 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         39 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         39 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         39 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         39 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               37 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         a 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       
                         a 
                         37 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         37 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         37 
                       
                     
                   
                   
                     
                       d 
                       37 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 40 connects the Proximal end of the Right Tibia to the Proximal end of the Right Fibula to form the Proximal Tibiofibular joint. The d 40  link length of the Right Proximal Tibia runs collinear with the Right Proximal Tibia running from the Proximal Tibia to the Right Proximal Fibula along the Y axis. 
     
       
         
           
             
               A 
               40 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               40 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               40 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               40 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               40 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             40 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           40 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               40 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       
                         a 
                         40 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       
                         a 
                         40 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         40 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         40 
                       
                     
                   
                   
                     
                       d 
                       40 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 41 connects the Proximal end of the Tibia with the Distal end of the Femur to create the Right Femorotibial joint. Normal range of motion (ROM) at the knee is considered to be zero (0) degrees of extension (completely straight knee joint) to 135 degrees of flexion (fully bent knee joint). The knee joint is one of the strongest and most important joints in the human body. It enables the lower leg to move relative to the femur while supporting the body&#39;s weight. Movements at the knee joint are essential to walking, running, sitting and standing. 
     The knee is a synovial hinge joint formed between three bones: The Femur, Tibia, and Patella. Two rounded, convex processes (known as condyles) on the Distal end of the Femur meet two rounded, concave condyles at the Proximal end of the Tibia. A special characteristic of the knee that differentiates it from other hinge joints is that it allows a small degree of medial and lateral rotation when it is moderately flexed. 
     
       
         
           
             
               A 
               41 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               41 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               41 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               41 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               41 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             41 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           41 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               41 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       
                         a 
                         41 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       
                         a 
                         41 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         41 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         41 
                       
                     
                   
                   
                     
                       d 
                       41 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 42 connects the Right femur Proximal end to the Right pelvis socket, forming the Right hip joint. 
     
       
         
           
             
               A 
               
                 4 
                  
                 2 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             
                               4 
                                
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         ψ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         ϕ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 4 
                  
                 2 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           4 
                            
                           2 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           4 
                            
                           2 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           2 
                         
                       
                     
                   
                   
                     
                       d 
                       
                         4 
                          
                         2 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 43 connects the Left Foot, 1 st  Toe, Hallux, Distal Phalange to the Left Foot, 1 st  Toe, Proximal Phalange. The d 43  link length of the Hallux Distal Phalange runs collinear with the Hallux centroid running from the Hallux base to the tip of the 1 st  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               43 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               43 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               43 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               43 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               43 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             43 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           43 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               43 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       
                         a 
                         43 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       
                         a 
                         43 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         43 
                       
                     
                   
                   
                     
                       d 
                       43 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 44 connects the Left Foot, 2 nd  Toe, Distal Phalange to the Left Foot 2 nd  toe Middle Phalange. The d 44  link length of the 2 nd  Distal Phalange runs collinear with the Distal Phalange centroid running from the 2 nd  Distal Phalange base to the tip of the 2 nd  Middle Phalange along the X axis. 
     
       
         
           
             
               A 
               
                 4 
                  
                 4 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 4 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 4 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 4 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               4 
                                
                               4 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             4 
                              
                             4 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           4 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           4 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           4 
                            
                           4 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 45 connects the Left foot, 3 rd  toe, Distal Phalange to the Left foot, 3 rd  toe Middle Phalange. The d 45  link length of the 3 rd  Distal Phalange runs collinear with the 3 rd  Distal Phalange centroid running from the 3 rd  Distal Phalange base to the tip of the 3 rd  Middle Phalange along the X axis. 
     
       
         
           
             
               A 
               
                 4 
                  
                 5 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               45 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               45 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               4 
                                
                               5 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             4 
                              
                             5 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   4 
                                    
                                   5 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 45 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 45 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   4 
                                    
                                   5 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       
                         4 
                          
                         5 
                       
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               a 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               a 
                               
                                 4 
                                  
                                 5 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 4 
                                  
                                 5 
                               
                             
                           
                         
                         
                           
                             d 
                             
                               4 
                                
                               5 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 46 connects the Left foot, 4 th  toe, Distal Phalange to the Left foot, 4 th  toe, Middle Phalange. The d 46  link length of the 4 th  Distal Phalange runs collinear with the 4 th  Distal Phalange centroid running from the 4 th  Distal Phalange base to the tip of the 4 th  Middle Phalange along the X axis. 
     
       
         
           
             
               A 
               46 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             46 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           46 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 46 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 46 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 46 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 46 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       46 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             
                               a 
                               46 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             
                               a 
                               46 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               46 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               46 
                             
                           
                         
                         
                           
                             d 
                             46 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 47 connects the Left foot, 5 th  toe, Distal Phalange to the Left foot, 5 th  toe, Middle Phalange. The d 47  link length of the 5 th  Distal Phalange runs collinear with the 4 th  Distal Phalange centroid running from the 5 th  Distal Phalange base to the tip of the 5 th  Middle Phalange along the X axis. 
     
       
         
           
             
               A 
               47 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               47 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               47 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               47 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               47 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             47 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           47 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               47 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       
                         a 
                         47 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       
                         a 
                         47 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         47 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         47 
                       
                     
                   
                   
                     
                       d 
                       47 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 48 connects the Left foot, 2 nd  toe, Middle Phalange to the Left foot, 2 nd  toe, Proximal Phalange. The d 48  link length of the 2 nd  Middle Phalange runs collinear with the 2 nd  Proximal Phalange centroid running from the 2 nd  Middle Phalange base to the tip of the 2 nd  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               48 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               48 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               48 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               48 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               48 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             48 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           48 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               48 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       
                         a 
                         48 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       
                         a 
                         48 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         48 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         48 
                       
                     
                   
                   
                     
                       d 
                       48 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 49 connects the Left foot, 3 rd  toe, Middle Phalange to the Left foot, 3 rd  toe, Proximal Phalange. The d 49  link length of the 3 rd  Middle Phalange runs collinear with the 3 rd  Middle Phalange centroid running from the 3 rd  Middle Phalange base to the tip of the 3 rd  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               49 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               49 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               49 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               49 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               49 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             49 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           49 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               49 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       
                         a 
                         49 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       
                         a 
                         49 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         49 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         49 
                       
                     
                   
                   
                     
                       d 
                       49 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 50 connects the Left foot, 4 th  toe, Middle Phalange to the Left foot, 4 th  toe, Proximal Phalange. The d 8  link length of the 4 th  Middle Phalange runs collinear with the 4 th  Middle Phalange centroid running from the 4 th  Middle Phalange base to the tip of the 4 th  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               50 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               50 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               50 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               50 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               50 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             50 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           50 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               50 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       
                         a 
                         50 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         43 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       
                         a 
                         50 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         50 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         50 
                       
                     
                   
                   
                     
                       d 
                       50 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 51 connects the Left foot, 5 th  toe, Middle Phalange to the Left foot, 5 th  toe Proximal Phalange. The d 51  link length of the 5 th  Middle Phalange runs collinear with the 5 th  Proximal Phalange centroid running from the 5 th  Middle Phalange base to the tip of the 5 th  Proximal Phalange along the X axis. 
     
       
         
           
             
               A 
               51 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               51 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               51 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               51 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               51 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             51 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           51 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       51 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               
                                 a 
                                 51 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               
                                 a 
                                 51 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 51 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 51 
                               
                             
                           
                           
                             
                               d 
                               51 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 52 connects the Left foot 1 st  toe Proximal Phalange to the Left foot, 1 st  toe Metatarsal. The d 52  link length of the 1 st  Proximal Phalange runs collinear with the 1 st  Proximal Phalange centroid running from the 1 st  Proximal Phalange base to the tip of the 1 st  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               52 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               52 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               52 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               52 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               52 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             52 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           52 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       52 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               
                                 a 
                                 52 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               
                                 a 
                                 52 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 52 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 52 
                               
                             
                           
                           
                             
                               d 
                               52 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 53 connects the Left foot, 2 nd  toe, Proximal Phalange to the Left foot, 2 nd  toe Metatarsal. The d 53  link length of the 2 nd  Proximal Phalange runs collinear with the 2 nd  Proximal Phalange centroid running from the 2 nd  Proximal Phalange base to the tip of the 2 nd  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               53 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               53 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               53 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               53 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               53 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             53 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           53 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       53 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               
                                 a 
                                 53 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               
                                 a 
                                 53 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 53 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 53 
                               
                             
                           
                           
                             
                               d 
                               53 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 54 connects the Left foot, 3 rd  toe, Proximal Phalange to the Left foot, 3 rd  toe Metatarsal. The d 54  link length of the 3 rd  Proximal Phalange runs collinear with the 3 rd  Proximal Phalange centroid running from the 3 rd  Proximal Phalange base to the tip of the 3 rd  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               54 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               54 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               54 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               54 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               54 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             54 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           54 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       54 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               
                                 a 
                                 54 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               
                                 a 
                                 54 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 54 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 54 
                               
                             
                           
                           
                             
                               d 
                               54 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 55 connects the Left foot, 4 th  toe Proximal Phalange to the Left foot 4 th  toe Metatarsal. The d 55  link length of the 4 th  Proximal Phalange runs collinear with the 4 th  Proximal Phalange centroid running from the 4 th  Proximal Phalange base to the tip of the 4 th  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               55 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               55 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               55 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               55 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               55 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             55 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           55 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       55 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               
                                 a 
                                 55 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               
                                 a 
                                 55 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 55 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 55 
                               
                             
                           
                           
                             
                               d 
                               55 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 56 connects the Left foot, 5 th  toe Proximal Phalange to the Left foot 5 th  toe Metatarsal. The d 56  link length of the 5 th  Proximal Phalange runs collinear with the 5 th  Proximal Phalange centroid running from the 5 th  Proximal Phalange base to the tip of the 5 th  Metatarsal along the X axis. 
     
       
         
           
             
               A 
               56 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               56 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               56 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               56 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               56 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             56 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           56 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       56 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               
                                 a 
                                 56 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               
                                 a 
                                 56 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 56 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 56 
                               
                             
                           
                           
                             
                               d 
                               56 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 57 connects the Left foot, 1 st  toe Metatarsal to the Left foot, 2 nd  toe Metatarsal. The d 57  link length of the Left big toe Metatarsal runs collinear with the 1 st  Metatarsal centroid running from the 1 st  Metatarsal to the 2 nd  Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               57 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               57 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               57 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               57 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               57 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             57 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           57 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       57 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               
                                 a 
                                 57 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               
                                 a 
                                 57 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 57 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 57 
                               
                             
                           
                           
                             
                               d 
                               57 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 58 connects the Left foot, 2 nd  toe Metatarsal to the Left foot 3 rd  toe Metatarsal. The d 23  link length of the Left 2 nd  Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal to the 3 rd  Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               58 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               58 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               58 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               58 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               58 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             58 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           58 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       58 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               
                                 a 
                                 58 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               
                                 a 
                                 58 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 58 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 58 
                               
                             
                           
                           
                             
                               d 
                               58 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 59 connects the Left foot, 3 rd  toe Metatarsal to the Left foot, 4 th  toe Metatarsal. The d 59  link length of the Left 4 th  Metatarsal runs collinear with the 3 rd  Metatarsal centroid running from the 3 rd  Metatarsal to the 3 rd  Metatarsal along the X axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               59 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               59 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               59 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               59 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               59 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             59 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           59 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       59 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               
                                 a 
                                 59 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               
                                 a 
                                 59 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 59 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 59 
                               
                             
                           
                           
                             
                               d 
                               59 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 60 connects the Left foot, 4th toe Metatarsal to the Left foot 5 th  toe Metatarsal. The d 60  link length of the Left 4 th  Metatarsal runs collinear with the 4 th  Metatarsal centroid running from the 4 th  Metatarsal to the 5 th  Metatarsal along the Y axis. This is a touch joint of adjacent metatarsals. 
     
       
         
           
             
               A 
               60 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               60 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               60 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               60 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               60 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             60 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           60 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       60 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               
                                 a 
                                 60 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               
                                 a 
                                 60 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 60 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 60 
                               
                             
                           
                           
                             
                               d 
                               60 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 61 connects the Left foot, 1 st  toe Metatarsal to the Left foot Medial Cuneiform. The d 61  link length of the Left big toe Metatarsal runs collinear with the 1 st  Metatarsal centroid running from the 1 st  Metatarsal base to the tip of the 1 st  Medial Cuneiform along the X axis. 
     
       
         
           
             
               A 
               61 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               61 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               61 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               61 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               61 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             61 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           61 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       61 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               
                                 a 
                                 61 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               
                                 a 
                                 61 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 61 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 61 
                               
                             
                           
                           
                             
                               d 
                               61 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 62 connects the Left foot, 2 nd  toe Metatarsal to the Left foot Medial Cuneiform. The d 62  link length of the Left big toe Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal base to the tip of the 2 nd  Medial Cuneiform along the X axis. 
     
       
         
           
             
               A 
               62 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               62 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               62 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               62 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               62 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             62 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           62 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       62 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               
                                 a 
                                 62 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               
                                 a 
                                 62 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 62 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 62 
                               
                             
                           
                           
                             
                               d 
                               62 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 63 connects the Left foot, 2 nd  toe Metatarsal to the Left foot Intermediate Cuneiform. The d 63  link length of the Left 2 nd  Metatarsal runs collinear with the Left Intermediate Cuneiform centroid running from the 2 nd  Metatarsal base to the tip of the Left Intermediate Cuneiform along the X axis 
     
       
         
           
             
               A 
               63 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               63 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               63 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               63 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               63 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             63 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           63 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       63 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               
                                 a 
                                 63 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               
                                 a 
                                 63 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 63 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 63 
                               
                             
                           
                           
                             
                               d 
                               63 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 64 connects the Left foot, 2 nd  toe Metatarsal to the Left foot Lateral Cuneiform. The d 64  link length of the Left big toe Metatarsal runs collinear with the 2 nd  Metatarsal centroid running from the 2 nd  Metatarsal base to the tip of the Left Lateral Cuneiform along the X axis. 
     
       
         
           
             
               A 
               64 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               64 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               64 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               64 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               64 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             64 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           64 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       64 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               
                                 a 
                                 64 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               
                                 a 
                                 64 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 64 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 64 
                               
                             
                           
                           
                             
                               d 
                               64 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 65 connects the Left foot, 3 rd  toe Metatarsal to the Left foot Lateral Cuneiform. The d 65  link length of the Left 3 rd  Metatarsal runs collinear with the 3 rd  Metatarsal centroid running from the 3 rd  Metatarsal base to the tip of the 3 rd  Lateral Cuneiform along the X axis. 
     
       
         
           
             
               A 
               65 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               65 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               65 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               65 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               65 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             65 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           65 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       65 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               
                                 a 
                                 65 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               
                                 a 
                                 65 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 65 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 65 
                               
                             
                           
                           
                             
                               d 
                               65 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 66 connects the left foot, 4 th  toe Metatarsal to the left fool Lateral Cuneiform. The d 66  link length of the left 4 th  Metatarsal runs collinear with the 4 th  Metatarsal centroid running from the 4 th  Metatarsal base to the tip of the Lateral Cuneiform along the X axis. 
     
       
         
           
             
               A 
               66 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               66 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               66 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               66 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               66 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             66 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           66 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       66 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               
                                 a 
                                 66 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               
                                 a 
                                 66 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 66 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 66 
                               
                             
                           
                           
                             
                               d 
                               66 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 67 connects the Left foot, 4th toe Metatarsal to the Left foot Cuboid bone. The d 67  link length of the Left 4 th  Metatarsal runs collinear with the 4th Metatarsal centroid running from the 4 th  Metatarsal base to the tip of the Left Cuboid along the X axis. 
     
       
         
           
             
               A 
               67 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               67 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               67 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               67 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               67 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             67 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           67 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       67 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               
                                 a 
                                 67 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               
                                 a 
                                 67 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 67 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 67 
                               
                             
                           
                           
                             
                               d 
                               67 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 68 connects the Left foot, 5th toe Metatarsal to the Left foot Cuboid bone. The d 68  link length of the Left 5th Metatarsal runs collinear with the 5th Metatarsal centroid running from the 5 th  Metatarsal base to the tip of the Left Cuboid along the X axis. 
     
       
         
           
             
               A 
               68 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               68 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               68 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               68 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               68 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             68 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           68 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       68 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               
                                 a 
                                 68 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               
                                 a 
                                 68 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 68 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 68 
                               
                             
                           
                           
                             
                               d 
                               68 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 69 connects the Left foot, Medial Cuneiform bone to the Left foot Intermediate Cuneiform bone. The d 69  link length of the Left Medial Cuneiform runs collinear with the Left Medial Cuneiform running from the Medial Cuneiform to the Left Intermediate Cuneiform along the Y axis. 
     
       
         
           
             
               A 
               69 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               69 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               69 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               69 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               69 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             69 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           69 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       69 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               
                                 a 
                                 69 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               
                                 a 
                                 69 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 69 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 69 
                               
                             
                           
                           
                             
                               d 
                               69 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 70 connects the left foot, Intermediate Cuneiform to the Left foot Lateral Cuneiform. The d 70  link length of the left Intermediate Cuneiform runs collinear with left Intermediate Cuneiform running from the Left Intermediate Cuneiform to the Lateral Cuneiform along the Y axis. 
     
       
         
           
             
               A 
               70 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               70 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               70 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               70 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               70 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             70 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           70 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       70 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               
                                 a 
                                 70 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               
                                 a 
                                 70 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 70 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 70 
                               
                             
                           
                           
                             
                               d 
                               70 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 71 connects the Left foot, Lateral Cuneiform to the left foot Cuboid. The d 71  link length of the Left Lateral Cuneiform runs collinear with the Left Lateral Cuneiform centroid running from the Left Lateral Cuneiform base to the tip of the Left Cuboid along the X axis. 
     
       
         
           
             
               A 
               71 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               71 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               71 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               71 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               71 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             71 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           71 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       71 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               
                                 a 
                                 71 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               
                                 a 
                                 71 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 71 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 71 
                               
                             
                           
                           
                             
                               d 
                               71 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 72 connects the Left foot, Medial Cuneiform bone to the Left foot Navicular bone. The d 70  link length of the Left Medial Cuneiform runs collinear with the Left Medial Cuneiform running from the Left Medial Cuneiform base to the tip of the Navicular along the X axis. 
     
       
         
           
             
               A 
               72 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               72 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               72 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               72 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               72 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             72 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           72 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       72 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               
                                 a 
                                 72 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               
                                 a 
                                 72 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 72 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 72 
                               
                             
                           
                           
                             
                               d 
                               72 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 73 connects the Left foot, Intermediate Cuneiform bone to the Left foot Navicular bone. This joint is both rotational of the Left Intermediate Cuneiform bone about the Left Navicular. The d 73  link length of the Left Intermediate Cuneiform runs collinear with the Left Intermediate Cuneiform running from the Left Intermediate Cuneiform base to the tip of the Left Navicular along the X axis. 
     
       
         
           
             
               A 
               73 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               73 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               73 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               73 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               73 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             73 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           73 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       73 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               
                                 a 
                                 73 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               
                                 a 
                                 73 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 73 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 73 
                               
                             
                           
                           
                             
                               d 
                               73 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 74 connects the Left foot, lateral Cuneiform bone to the Left foot Navicular bone. This joint is both rotational of the Left lateral Cuneiform bone about the Left Navicular in the X plane, and rotation of the Left lateral Cuneiform bone about the Left Navicular in the Y plane, in an oblong interface supporting rotational motion in two direction. There is no linear motion in either the X plane or the Y plane in this joint. 
     
       
         
           
             
               A 
               74 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               74 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               74 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               74 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               74 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             74 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           74 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       74 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               
                                 a 
                                 74 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               
                                 a 
                                 74 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 74 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 74 
                               
                             
                           
                           
                             
                               d 
                               74 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 75 connects the Left foot, Navicular to the Left foot Cuboid. The d 75  link length of the Left Navicular runs collinear with the Left Navicular centroid running from the Left Navicular to the Cuboid along the Y axis. 
     
       
         
           
             
               A 
               75 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               75 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               75 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               75 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               75 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             75 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           75 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       75 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               
                                 a 
                                 75 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               
                                 a 
                                 75 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 75 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 75 
                               
                             
                           
                           
                             
                               d 
                               75 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 76 connects the Left foot, Navicular bone to the Left foot Talus bone. This joint is a rotational joint with respect to the Navicular bone and the Talus bone in the Z plane and is also a rotational joint with respect to the Navicular bone and the Talus bone in the Y plane. There is no linear lateral motion in this joint, rather, the curved structure of the Talus-Navicular interface is curved, thus supporting only rotation. 
     
       
         
           
             
               A 
               76 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               76 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               76 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               76 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               76 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             76 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           76 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       76 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               
                                 a 
                                 76 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               
                                 a 
                                 76 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 76 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 76 
                               
                             
                           
                           
                             
                               d 
                               76 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 77 connects the Left foot, Cuboid to the Left foot Calcaneus. The d 77  link length of the Left Calcaneus runs collinear with the Left Cuboid running from the Left Cuboid base to the tip of the Left Calcaneus along the X axis. 
     
       
         
           
             
               A 
               77 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               77 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               77 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               77 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               77 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             77 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           77 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       77 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               
                                 a 
                                 77 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               
                                 a 
                                 77 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 77 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 77 
                               
                             
                           
                           
                             
                               d 
                               77 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 78 connects the Left foot, Talus to the left foot Calcaneus. The d 78  link length of the left Talus runs collinear with the left Talus running from the Talus to the Calcaneus along the Y axis. 
     
       
         
           
             
               A 
               78 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               78 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               78 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               78 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               78 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             78 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           78 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       78 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               
                                 a 
                                 78 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               
                                 a 
                                 78 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 78 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 78 
                               
                             
                           
                           
                             
                               d 
                               78 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 79 connects the Left Fibula to the Left Talus bone, this is commonly referred to as the ankle joint. The d 79  link length of the Left Fibula runs collinear with the Left Fibula running from the Fibula base to the Talus along the X axis. 
     
       
         
           
             
               A 
               79 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               79 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               79 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               79 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               79 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             79 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           79 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       79 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               
                                 a 
                                 79 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               
                                 a 
                                 79 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 79 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 79 
                               
                             
                           
                           
                             
                               d 
                               79 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 80 connects the Left tibia to the Left talus bone. The d 80  link length of the Left Tibia runs collinear with the Left Tibia running from the Fibula base to the Talus along the X axis. 
     
       
         
           
             
               A 
               80 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               80 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               80 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               80 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               80 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             80 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           80 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       80 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               
                                 a 
                                 80 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               
                                 a 
                                 80 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 80 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 80 
                               
                             
                           
                           
                             
                               d 
                               80 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 81 connects the Dorsal end of the Left Tibia to the Dorsal end of the Left Fibula to form the Dorsal Tibiofibular joint. The d 81  link length of the Left Doral Tibia runs collinear with the Left Dorsal Tibia running from the Dorsal Tibia to the Left Dorsal Fibula along the Y axis. 
     
       
         
           
             
               A 
               81 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               81 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               81 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               81 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               81 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             81 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             81 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           81 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       81 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               
                                 a 
                                 81 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               
                                 a 
                                 81 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 81 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 81 
                               
                             
                           
                           
                             
                               d 
                               81 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 82 connects the Proximal end of the Left Tibia to the Proximal end of the Left Fibula to form the Proximal Tibiofibular joint. The d 82  link length of the Left Proximal Tibia runs collinear with the Left Proximal Tibia running from the Proximal Tibia to the Left Proximal Fibula along the Y axis. 
     
       
         
           
             
               A 
               82 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               82 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               82 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               82 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               82 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             82 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             82 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           82 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       82 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               
                                 a 
                                 82 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               
                                 a 
                                 82 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 82 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 82 
                               
                             
                           
                           
                             
                               d 
                               82 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 83 connects the Proximal end of the Tibia with the Distal end of the Femur to create the Left Femorotibial joint. Normal range of motion (ROM) at the knee is considered to be zero (0) degrees of extension (completely straight knee joint) to 135 degrees of flexion (fully bent knee joint). The knee joint is one of the strongest and most important joints in the human body. It enables the lower leg to move relative to the femur while supporting the body&#39;s weight. Movements at the knee joint are essential to walking, running, sitting and standing. The knee is a synovial hinge joint formed between three bones: the Femur, Tibia, and Patella. Two rounded convex processes (known as condyles) on the Distal end of the Femur meet two rounded, concave condyles at the Proximal end of the Tibia. A special characteristic of the knee that differentiates it from other hinge joints is that it allows a small degree of medial and lateral rotation when it is moderately flexed. 
     
       
         
           
             
               A 
               83 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               83 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               83 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               83 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               83 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             83 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             83 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           83 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       83 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               
                                 a 
                                 83 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               
                                 a 
                                 83 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 83 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 83 
                               
                             
                           
                           
                             
                               d 
                               83 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 84 connects the Left femur Proximal end to the Left pelvis socket, forming the Left hip joint. 
     
       
         
           
             
               A 
               84 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             84 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             84 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             84 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             84 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           84 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             84 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             84 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             84 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             84 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 84 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 84 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 84 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 84 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       84 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               
                                 a 
                                 84 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               
                                 a 
                                 84 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 84 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 84 
                               
                             
                           
                           
                             
                               d 
                               84 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 85 connects the 0 th  Cervical Vertebra to the 1 st  Cervical Vertebra and is known as the occipito-atlanto joint. The C1 Vertebra is also known as the Atlas. The occipito-atlanto joint is a synovial joint whose main purpose is to bend forward and backward. It is formed by paired occipital condyles and superior articular facets of the atlas. The d 85  link length of the C0 vertebra runs collinear with the C0 vertebra running from the head of the C1 vertebra to the base of the skull along the X axis. 
     
       
         
           
             
               A 
               85 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             85 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             85 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             85 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             85 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           85 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             85 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             85 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             85 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             85 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 85 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 85 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 85 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 85 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       85 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               
                                 a 
                                 85 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               
                                 a 
                                 85 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 85 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 85 
                               
                             
                           
                           
                             
                               d 
                               85 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 86 connects the 1 st  Cervical Vertebra to the 2 nd  Cervical Vertebra. This is also termed the Axis. 50% of the rotation enabled by the C0 through C7 vertebra is realized by this joint alone. The range of rotation is 50 degrees to each side. The d 86  link length of the C1 vertebra runs collinear with the C1 vertebra running from the top of the C2 vertebra to the base of the C1 vertebra along the X axis. 
     
       
         
           
             
               A 
               86 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             86 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             86 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             86 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             86 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           
                             d 
                             86 
                           
                            
                           0 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             86 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             86 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             86 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             86 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 86 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 86 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 86 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 86 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       86 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               
                                 a 
                                 86 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               
                                 a 
                                 86 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 86 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 86 
                               
                             
                           
                           
                             
                               d 
                               86 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 87 connects the 2 nd  Cervical Vertebra to the 3 rd  Cervical Vertebra. The C2-C3 joint provides movement of the chin toward the chest (flexion) and the backward movement of the head (extension). The d 87  link length of the C2 vertebra runs collinear with the C2 vertebra running from the top of the C3 vertebra to the base of the C2 vertebra along the X axis. 
     
       
         
           
             
               A 
               87 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             87 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             87 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             87 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             87 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           87 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             87 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             87 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             87 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             87 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 87 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 87 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 87 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 87 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       87 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               
                                 a 
                                 87 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               
                                 a 
                                 87 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 87 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 87 
                               
                             
                           
                           
                             
                               d 
                               87 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 88 connects the 3 rd  Cervical Vertebra to the 4 th  Cervical Vertebra. The d 88  link length of the C3 vertebra runs collinear with the C3 vertebra running from the head of the C4 vertebra to the base of the C3 vertebra along the X axis. 
     
       
         
           
             
               A 
               88 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             88 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             88 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             88 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             88 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           88 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             88 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             88 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             88 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             88 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 88 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 88 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 88 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 88 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       88 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               
                                 a 
                                 88 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               
                                 a 
                                 88 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 88 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 88 
                               
                             
                           
                           
                             
                               d 
                               88 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 89 connects the 4 th  cervical vertebra to the 5 th  cervical vertebra. The d 89  link length of the C3 vertebra runs collinear with the C0 vertebra running from the head of the C5 vertebra to the base of the C4 vertebra along the X axis. 
     
       
         
           
             
               A 
               89 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             89 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             89 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             89 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             89 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           89 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             89 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             89 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             89 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             89 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 89 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 89 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 89 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 89 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       89 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               
                                 a 
                                 89 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               
                                 a 
                                 89 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 89 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 89 
                               
                             
                           
                           
                             
                               d 
                               89 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 90 connects the 5 th  Cervical Vertebra to the 6 th  Cervical Vertebra. The d 90  link length of the C5 vertebra runs collinear with the C5 vertebra running from the head of the C6 vertebra to the base of the C5 vertebra along the X axis. 
     
       
         
           
             
               A 
               90 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             90 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             90 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             90 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             90 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           90 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             90 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             90 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             90 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             90 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 90 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 90 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 90 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 90 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       90 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               
                                 a 
                                 90 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               
                                 a 
                                 90 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 90 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 90 
                               
                             
                           
                           
                             
                               d 
                               90 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 91 connects the 6th Cervical Vertebra to the 7th Cervical Vertebra. The d 91  link length of the C6 vertebra runs collinear with the C6 vertebra running from the head of the C7 vertebra to the base of the C6 vertebra along the X axis. 
     
       
         
           
             
               A 
               91 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             91 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             91 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             91 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             91 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           91 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             91 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             91 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             91 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             91 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 91 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 91 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 91 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 91 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       91 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               
                                 a 
                                 91 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               
                                 a 
                                 91 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 91 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 91 
                               
                             
                           
                           
                             
                               d 
                               91 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 92 connects the 7 th  cervical vertebra to the 1 st  thoracic vertebra. The d 92  link length of the C7 vertebra runs collinear with the C7 vertebra running from the head of the T1 vertebra to the base of the C7 vertebra along the X axis. 
     
       
         
           
             
               A 
               92 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             92 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             92 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             92 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             92 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           92 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             92 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             92 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             92 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             92 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 92 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 92 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 92 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 92 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       92 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               
                                 a 
                                 92 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               
                                 a 
                                 92 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 92 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 92 
                               
                             
                           
                           
                             
                               d 
                               92 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 93 connects the 1 st  Thoracic Vertebra to the 2nd Thoracic Vertebra. The d 93  link length of the T1 vertebra runs collinear with the T1 vertebra running from the head of the T2 vertebra to the base of the T1 vertebra along the X axis. 
     
       
         
           
             
               A 
               93 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             93 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             93 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             93 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             93 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           93 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             93 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             93 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             93 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             93 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 93 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 93 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 93 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 93 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       93 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               
                                 a 
                                 93 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               
                                 a 
                                 93 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 93 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 93 
                               
                             
                           
                           
                             
                               d 
                               93 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 94 connects the 2 nd  Thoracic Vertebra to the 3rd Thoracic Vertebra. The d 94  link length of the T2 vertebra runs collinear with the T2 vertebra running from the head of the T3 vertebra to the base of the T2 vertebra along the X axis. 
     
       
         
           
             
               A 
               94 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             94 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             94 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             94 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             94 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           94 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             94 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             94 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             94 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             94 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 94 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 94 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 94 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 94 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       94 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               
                                 a 
                                 94 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               
                                 a 
                                 94 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 94 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 94 
                               
                             
                           
                           
                             
                               d 
                               94 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 95 connects the 3 rd  Thoracic Vertebra to the 4th Thoracic Vertebra. The d 95  link length of the T3 vertebra runs collinear with the T3 vertebra running from the head of the T4 vertebra to the base of the T3 vertebra along the X axis. 
     
       
         
           
             
               A 
               95 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             95 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             95 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             95 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             95 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           95 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             95 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             95 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             95 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             95 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 95 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 95 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 95 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 95 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       95 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               
                                 a 
                                 95 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               
                                 a 
                                 95 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 95 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 95 
                               
                             
                           
                           
                             
                               d 
                               95 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 96 connects the 4 th  thoracic vertebra to the 5 th  thoracic vertebra. The d 96  link length of the T4 vertebra runs collinear with the T4 vertebra running from the head of the T5 vertebra to the base of the T4 vertebra along the X axis. 
     
       
         
           
             
               A 
               96 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           96 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             96 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 96 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 96 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 96 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 96 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       96 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               
                                 a 
                                 96 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               
                                 a 
                                 96 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 96 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               d 
                               96 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 97 connects the 5 th  Thoracic Vertebra to the 6 th  Thoracic Vertebra. The d 97  link length of the T5 vertebra runs collinear with the T5 vertebra running from the head of the T6 vertebra to the base of the T5 vertebra along the X axis. 
     
       
         
           
             
               A 
               97 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           97 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             97 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 97 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 97 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 97 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 97 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       97 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 97 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 97 
                               
                             
                           
                           
                             
                               
                                 a 
                                 96 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 97 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 97 
                               
                             
                           
                           
                             
                               
                                 a 
                                 96 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 97 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 96 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 97 
                               
                             
                           
                           
                             
                               d 
                               97 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 98 connects the 6 th  Thoracic Vertebra to the 7 th  Thoracic Vertebra. The d 98  link length of the T6 vertebra runs collinear with the T6 vertebra running from the head of the T7 vertebra to the base of the T6 vertebra along the X axis. 
     
       
         
           
             
               A 
               98 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           98 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             98 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 98 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 98 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 98 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 98 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       98 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               
                                 a 
                                 98 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               
                                 a 
                                 98 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 98 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 98 
                               
                             
                           
                           
                             
                               d 
                               98 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 99 connects the 7 th  Thoracic Vertebra to the 8 th  Thoracic Vertebra. The d 99  link length of the T7 vertebra runs collinear with the T7 vertebra running from the head of the T8 vertebra to the base of the T7 vertebra along the X axis. 
     
       
         
           
             
               A 
               99 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           99 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             99 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 99 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 99 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 99 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 99 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       99 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               
                                 a 
                                 99 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               
                                 a 
                                 99 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 99 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 99 
                               
                             
                           
                           
                             
                               d 
                               99 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 100 connects the 8 th  Thoracic Vertebra to the 9 th  Thoracic Vertebra. The d 100  link length of the T8 vertebra runs collinear with the T8 vertebra running from the head of the T9 vertebra to the base of the T8 vertebra along the X axis. 
     
       
         
           
             
               A 
               100 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           100 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             100 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 100 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 100 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 100 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 100 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       100 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               
                                 a 
                                 100 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               
                                 a 
                                 100 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 100 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 100 
                               
                             
                           
                           
                             
                               d 
                               100 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 101 connects the 9 th  Thoracic Vertebra to the 10 th  Thoracic Vertebra. The d 101  link length of the T8 vertebra runs collinear with the T8 vertebra running from the head of the T9 vertebra to the base of the T8 vertebra along the X axis. 
     
       
         
           
             
               A 
               101 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           101 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             101 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 101 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 101 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 101 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 101 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       101 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               
                                 a 
                                 101 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               
                                 a 
                                 101 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 101 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 101 
                               
                             
                           
                           
                             
                               d 
                               101 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 102 connects the 10th Thoracic Vertebra to the 11 th  Thoracic Vertebra. The d 98  link length of the T10 vertebra runs collinear with the T10 vertebra running from the head of the T11 vertebra to the base of the T10 vertebra along the X axis. 
     
       
         
           
             
               A 
               102 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           102 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             102 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 102 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 102 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 102 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 102 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       102 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               
                                 a 
                                 102 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               
                                 a 
                                 102 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 102 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 102 
                               
                             
                           
                           
                             
                               d 
                               102 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 103 connects the 11th Thoracic Vertebra to the 12 th  Thoracic Vertebra. The d 103  link length of the T11 vertebra runs collinear with the T11 vertebra running from the head of the T12 vertebra to the base of the T11 vertebra along the X axis. 
     
       
         
           
             
               A 
               103 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           103 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             103 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 103 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 103 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 103 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 103 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       103 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               
                                 a 
                                 103 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               
                                 a 
                                 103 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 103 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 103 
                               
                             
                           
                           
                             
                               d 
                               103 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 104 connects the 12th Thoracic Vertebra to the 1 st  Lumbar Vertebra. The d 104  link length of the T12 vertebra runs collinear with the T12 vertebra running from the head of the T12 vertebra to the base of the L1 vertebra along the X axis. 
     
       
         
           
             
               A 
               104 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           104 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             104 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 104 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 104 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 104 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 104 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       104 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               
                                 a 
                                 104 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               
                                 a 
                                 104 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 104 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 104 
                               
                             
                           
                           
                             
                               d 
                               104 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 105 connects the 1st lumbar vertebra to the 2 nd  lumbar vertebra. The d 105  link length of the L2 vertebra runs collinear with the L2 vertebra running from the head of the L2 vertebra to the base of the L1 vertebra along the X axis. 
     
       
         
           
             
               A 
               105 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           105 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             105 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 105 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 105 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 105 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 105 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       105 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               
                                 a 
                                 105 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               
                                 a 
                                 105 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 105 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 105 
                               
                             
                           
                           
                             
                               d 
                               105 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 106 connects the 2 nd  Lumbar Vertebra to the 3 rd  Lumbar Vertebra. The d 106  link length of the L2 vertebra runs collinear with the L2 vertebra running from the head of the L3 vertebra to the base of the L2 vertebra along the X axis. 
     
       
         
           
             
               A 
               106 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           106 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             106 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 106 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 106 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 106 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 106 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       106 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               
                                 a 
                                 106 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               
                                 a 
                                 106 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 106 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 106 
                               
                             
                           
                           
                             
                               d 
                               106 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 107 connects the 3 rd  Lumbar Vertebra to the 4 th  Lumbar Vertebra. The d 107  link length of the L3 vertebra runs collinear with the L3 vertebra running from the head of the L4 vertebra to the base of the L3 vertebra along the X axis. 
     
       
         
           
             
               A 
               107 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           107 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             107 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 107 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 107 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 107 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 107 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       107 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               
                                 a 
                                 107 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               
                                 a 
                                 107 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 107 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 107 
                               
                             
                           
                           
                             
                               d 
                               107 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 108 connects the 4 th  Lumbar Vertebra to the 5 th  Lumbar Vertebra. The d 108  link length of the L4 vertebra runs collinear with the L4 vertebra running from the head of the L5 vertebra to the base of the L4 vertebra along the X axis. 
     
       
         
           
             
               A 
               108 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           108 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             108 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 108 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 108 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 108 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 108 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       108 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               
                                 a 
                                 108 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               
                                 a 
                                 108 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 108 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 108 
                               
                             
                           
                           
                             
                               d 
                               108 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 109 connects the 5 th  Lumbar Vertebra to the 2 nd  Lumbar Vertebra. The d 109  link length of the L5 vertebra runs collinear with the L5 vertebra running from the head of the Si sacrum to the base of the L5 vertebra along the X axis. 
     
       
         
           
             
               A 
               109 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           109 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             109 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 109 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 109 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 109 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 109 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       109 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               
                                 a 
                                 109 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               
                                 a 
                                 109 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 109 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 109 
                               
                             
                           
                           
                             
                               d 
                               109 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 110 connects the right thumb Distal Phalanx to the right thumb Proximal Phalanx, forming the 1 st  distal interphalangeal joint (DIP). The duo length of the right Distal Phalanx runs through the centroid of the Distal Phalanx from the head of the right Proximal Phalanx to the head of the right distal phalanx 
     
       
         
           
             
               A 
               110 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           110 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             110 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ψ 
                                 110 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 110 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 110 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 110 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       110 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               
                                 a 
                                 110 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                             
                           
                         
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               
                                 - 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               
                                 a 
                                 110 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 110 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 110 
                               
                             
                           
                           
                             
                               d 
                               110 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Joint 111 connects the right index finger Distal Phalanx to the right index finger Middle Phalanx, forming the 2 nd  distal interphalangeal joint (DIP) The d ill  length of the right index finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the right index finger Proximal Phalanx to the head of the right index finger Distal Phalanx. 
     
       
         
           
             
               A 
               
                 1 
                  
                 1 
                  
                 1 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               1 
                                
                               1 
                                
                               1 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             1 
                              
                             1 
                              
                             1 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 111 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 111 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 111 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 111 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       111 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             
                               a 
                               
                                 1 
                                  
                                 1 
                                  
                                 1 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 1 
                                  
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             
                               a 
                               111 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               111 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               111 
                             
                           
                         
                         
                           
                             d 
                             111 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 112 connects the right middle finger Distal Phalanx to the right middle finger Middle Phalanx, forming the 3 rd  distal interphalangeal joint (DIP). The d 112  length of the right middle finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the right index finger Proximal Phalanx to the dead of the right middle finger Distal Phalanx. 
     
       
         
           
             
               A 
               112 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             112 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           112 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 112 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 112 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 112 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 112 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       112 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             
                               a 
                               112 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             
                               a 
                               112 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               112 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               112 
                             
                           
                         
                         
                           
                             d 
                             112 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 113 connects the right ring finger Distal Phalanx to the right ring finger Middle Phalanx, forming the 4 th  distal interphalangeal joint (DIP). The d 113  length of the right ring finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the right ring finger Proximal Phalanx to the head of the right ring finger Distal Phalanx. 
     
       
         
           
             
               A 
               113 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             113 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           113 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 113 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 113 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 113 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 113 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       113 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             
                               a 
                               113 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             
                               a 
                               113 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               113 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               113 
                             
                           
                         
                         
                           
                             d 
                             113 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 114 connects the right little finger Distal Phalanx to the right little finger Middle Phalanx, forming the 5 th  distal interphalangeal joint (DIP). The d 114  length of the right little finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the right little finger Proximal Phalanx to the head of the right little finger Distal Phalanx. 
     
       
         
           
             
               A 
               114 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             114 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           114 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 114 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 114 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 114 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 114 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       114 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             
                               a 
                               113 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             
                               a 
                               113 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               114 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               114 
                             
                           
                         
                         
                           
                             d 
                             114 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 115 connects the right index finger Middle Phalanx to the right index finger Proximal Phalanx forming the 2 nd  proximal interphalangeal joint (PIP). The d 115  length of the right index finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the right index finger Proximal Phalanx to the head of the right index finger middle phalanx. 
     
       
         
           
             
               A 
               115 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             115 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           115 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 115 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 115 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 115 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 115 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       115 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             
                               a 
                               115 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             
                               a 
                               115 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               115 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               115 
                             
                           
                         
                         
                           
                             d 
                             115 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 116 connects the right middle finger Middle Phalanx to the right middle finger Proximal Phalanx, forming the 3 rd  proximal interphalangeal joint (PIP). 
     The d 116  length of the right middle finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the right middle finger Proximal Phalanx to the head of the right middle finger Middle Phalanx. 
     
       
         
           
             
               A 
               116 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             116 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           116 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 116 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 116 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 116 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 116 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       116 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             
                               a 
                               116 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             
                               a 
                               116 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               116 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               116 
                             
                           
                         
                         
                           
                             d 
                             116 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 117 connects the right ring finger Middle Phalanx to the right ring finger Proximal Phalanx, forming the 4 th  proximal interphalangeal joint (PIP). The d 117  length of the right ring finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the right ring finger Proximal Phalanx to the head of the right ring finger Middle Phalanx. 
     
       
         
           
             
               A 
               117 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             117 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           117 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 117 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 117 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 117 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 117 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       117 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             
                               a 
                               117 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             
                               a 
                               117 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               117 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               117 
                             
                           
                         
                         
                           
                             d 
                             117 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 118 connects the right little finger Middle Phalanx to the right little finger Proximal Phalanx, forming the 5 th  proximal interphalangeal joint (PIP). The d 117  length of the right little finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the right little finger Proximal Phalanx to the head of the right little finger Middle Phalanx. 
     
       
         
           
             
               A 
               118 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             118 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           118 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 118 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 118 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 118 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 118 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       118 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             
                               a 
                               118 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             
                               a 
                               118 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               118 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               118 
                             
                           
                         
                         
                           
                             d 
                             118 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 119 connects the right thumb Proximal Phalanx to the right thumb Metacarpal in the 1 st  metacarpophalangeal joint (MCP). The d 119  length of the right thumb Middle Phalanx runs through the centroid of the Middle Proximal Phalanx from the head of the thumb Metacarpal to the head of the thumb Proximal Phalanx. 
     
       
         
           
             
               A 
               119 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             119 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           119 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 119 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 119 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 119 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 119 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       119 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             
                               a 
                               119 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             
                               a 
                               119 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               119 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               119 
                             
                           
                         
                         
                           
                             d 
                             119 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 120 connects the right index finger Proximal Phalanx to the right index finger Metacarpal in the 2 nd  metacarpophalangeal joint (MCP). The d 120  length of the right index finger Proximal Phalanx runs through the centroid of the index finger Proximal Phalanx from the head of the index finger Metacarpus to the head of the index finger Proximal Phalanx. 
     
       
         
           
             
               A 
               120 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             120 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           120 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 120 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 120 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 120 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 120 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       120 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             
                               a 
                               120 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             
                               a 
                               120 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               120 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               120 
                             
                           
                         
                         
                           
                             d 
                             120 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 121 connects the right middle finger Proximal Phalanx to the right middle finger Metacarpal in the 3 rd  metacarpophalangeal joint (MCP). The d 121  length of the right middle finger Proximal Phalanx runs through the centroid of the right middle finger Proximal Phalanx from the head of the middle finger Metacarpus to the head of the middle finger Proximal Phalanx. 
     
       
         
           
             
               A 
               121 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             121 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           121 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 121 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 121 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 121 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 121 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       121 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             
                               a 
                               121 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             
                               a 
                               121 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               121 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               121 
                             
                           
                         
                         
                           
                             d 
                             121 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 122 connects the right ring finger Proximal Phalanx to the right ring finger Metacarpal in the 4 th  metacarpophalangeal joint (MCP). The d 122  length of the right ring finger Proximal Phalanx runs through the centroid of the right ring finger Proximal Phalanx from the head of the ring finger Metacarpus to the head of the ring finger Proximal Phalanx. 
     
       
         
           
             
               A 
               122 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             122 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           122 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 122 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 122 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 122 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 122 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       122 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             
                               a 
                               122 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             
                               a 
                               122 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               122 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               122 
                             
                           
                         
                         
                           
                             d 
                             122 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 123 connects the right little finger Proximal Phalanx to the right little finger Metacarpal in the 5 th  metacarpophalangeal joint (MCP). The d 123  length of the right little finger Proximal Phalanx runs through the centroid of the right little finger Proximal Phalanx from the head of the little finger Metacarpus to the head of the little finger Proximal Phalanx. 
     
       
         
           
             
               A 
               123 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             123 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           123 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 123 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 123 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 123 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 123 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       123 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             
                               a 
                               123 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             
                               a 
                               123 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               123 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               123 
                             
                           
                         
                         
                           
                             d 
                             123 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 124 connects the right thumb Metacarpal to the right Trapezium. The d 124  length of the right thumb Metacarpal runs through the centroid of the right thumb Metacarpal. 
     
       
         
           
             
               A 
               124 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             124 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           124 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 124 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 124 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 124 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 124 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       124 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             
                               a 
                               124 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             
                               a 
                               124 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               124 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               124 
                             
                           
                         
                         
                           
                             d 
                             124 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 125 connects the right thumb Metacarpal to the right Index finger Metacarpal. The d 125  length of the right thumb Metacarpal runs through the centroid of the right thumb Metacarpal, beginning at the head of the right Trapezium, and extending to the head of the right thumb Metacarpal. 
     
       
         
           
             
               A 
               125 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             125 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           125 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 125 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 125 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 125 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 125 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       125 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             
                               a 
                               125 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             
                               a 
                               125 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               125 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               125 
                             
                           
                         
                         
                           
                             d 
                             125 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 126 connects the right index finger Metacarpal to the right middle finger Metacarpal. The d 126  length of the right index finger Metacarpal runs through the centroid of the right index finger Metacarpal beginning at the side of the right middle finger Metacarpal. 
     
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 6 
               
             
             = 
             
               
 
             
              
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 6 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 6 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           
                             1 
                              
                             2 
                              
                             6 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
                
               
                 
 
               
               [ 
               
                   
               
                
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 6 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         126 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         126 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         co 
                       
                        
                       
                           
                       
                        
                       s 
                        
                       
                           
                       
                        
                       
                         θ 
                         126 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           6 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         126 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         126 
                       
                     
                   
                   
                     
                       d 
                       
                         1 
                          
                         2 
                          
                         6 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 127 connects the right middle finger Metacarpal to the right ring finger Metacarpal. The d 127  length of the right middle finger Metacarpal runs through the right middle finger Metacarpal beginning at the side of the right middle finger Metacarpal to the head of the right middle finger Metacarpal. 
     
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 7 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 7 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 7 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 7 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 7 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               1 
                                
                               2 
                                
                               7 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             1 
                              
                             2 
                              
                             7 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 7 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         127 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         127 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         127 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           7 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         127 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         127 
                       
                     
                   
                   
                     
                       d 
                       127 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 128 connects the right index finger Metacarpal to the Right Trapezium. The d 128  length of the right index finger Metacarpal runs through the right index finger Metacarpal beginning at the head of the right trapezium to the head of the right index finger Metacarpal. 
     
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 8 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 8 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 8 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 8 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 8 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               1 
                                
                               2 
                                
                               8 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             1 
                              
                             2 
                              
                             8 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 8 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       si 
                        
                       
                           
                       
                        
                       n 
                        
                       
                           
                       
                        
                       
                         θ 
                         128 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         128 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         128 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           8 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         128 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         128 
                       
                     
                   
                   
                     
                       d 
                       
                         1 
                          
                         2 
                          
                         8 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 129 connects the right index finger Metacarpal to the right Trapezoid. The d 128  length of the right index finger Metacarpal runs through the right index finger Metacarpal beginning at the head of the right trapezoid to the head of the right index finger Metacarpal. 
     
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 9 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 9 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 9 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 9 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 2 
                                  
                                 9 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               1 
                                
                               2 
                                
                               9 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             1 
                              
                             2 
                              
                             9 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               
                 1 
                  
                 2 
                  
                 9 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         129 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         129 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         129 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                   
                     
                       
                         a 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         
                           1 
                            
                           2 
                            
                           9 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         129 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         129 
                       
                     
                   
                   
                     
                       d 
                       
                         1 
                          
                         2 
                          
                         9 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 130 connects the right middle finger to the right Trapezoid. The d 130  length of the right index finger Metacarpal runs through the right index finger Metacarpal beginning at the head of the right trapezoid to the head of the right index finger Metacarpal. 
     
       
         
           
             
               A 
               130 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               130 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               130 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               130 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               130 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             130 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           130 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               130 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       
                         a 
                         130 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       
                         a 
                         130 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         130 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         130 
                       
                     
                   
                   
                     
                       d 
                       130 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 131 connects the right middle finger Metacarpal to the right Capitate. The d 131  length of the right index finger Metacarpal runs through the right index finger Metacarpal beginning at the head of the right Capitate to the head of the right index finger Metacarpal. 
     
       
         
           
             
               A 
               131 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               131 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               131 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               131 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               131 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             131 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           131 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               131 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       
                         a 
                         131 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       
                         a 
                         131 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         131 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         131 
                       
                     
                   
                   
                     
                       d 
                       131 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 132 connects the right middle finger Metacarpal to the right ring finger Metacarpal. The d 132  length of the right middle finger Metacarpal runs through the right middle finger Metacarpal beginning at the head of the right Metacarpal to the head of the right middle finger Metacarpal. 
     
       
         
           
             
               A 
               132 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               132 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               132 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               132 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               132 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             132 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           132 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               132 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       
                         a 
                         132 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       
                         a 
                         132 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         132 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         132 
                       
                     
                   
                   
                     
                       d 
                       132 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 133 connects the right ring finger Metacarpal to the Right Capitate. The d 133  length of the right ring finger Metacarpal runs through the right middle finger Metacarpal beginning at the head of the right Metacarpal to the head of the right Capitate. 
     
       
         
           
             
               A 
               133 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               133 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               133 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               133 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               133 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             133 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           133 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               133 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       
                         a 
                         133 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       
                         a 
                         133 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         133 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         133 
                       
                     
                   
                   
                     
                       d 
                       133 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 134 connects the right ring finger Metacarpal to the right little finger Metacarpal. The d 134  length of the right ring finger Metacarpal runs through the right middle finger Metacarpal beginning at the touchpoint of the right little finger Metacarpal to the touchpoint on the little finger Metacarpal. 
     
       
         
           
             
               A 
               134 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               134 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               134 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               134 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               134 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             134 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           134 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               134 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       
                         a 
                         134 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       
                         a 
                         134 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         134 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         134 
                       
                     
                   
                   
                     
                       d 
                       134 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 135 connects the right ring finger Metacarpal to the right Hamate. The d 135  length of the right ring finger Metacarpal runs through the right middle finger Metacarpal beginning at the touchpoint of the right little finger Metacarpal to the touchpoint on the right Hamate. 
     
       
         
           
             
               A 
               135 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               135 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               135 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               135 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               135 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             135 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           135 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               135 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       
                         a 
                         135 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       
                         a 
                         135 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         135 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         135 
                       
                     
                   
                   
                     
                       d 
                       135 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 136 connects the right little finger Metacarpal to the right Hamate. The d 136  length of the right little finger Metacarpal runs through the right little finger Metacarpal beginning at the touchpoint of the right little finger Metacarpal to the touchpoint on the right Hamate. 
     
       
         
           
             
               A 
               136 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               136 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               136 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               136 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               136 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             136 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           136 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               136 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       
                         a 
                         136 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       
                         a 
                         136 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         136 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         136 
                       
                     
                   
                   
                     
                       d 
                       136 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 137 connects the right Trapezium to the right Trapezoid. The d 137  length of the right Trapezium runs through the right Trapezium beginning at the touchpoint of the right Trapezoid to the touchpoint on the right Trapezium. 
     
       
         
           
             
               A 
               137 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               137 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               137 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               137 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               137 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             137 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           137 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               137 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       
                         a 
                         137 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       
                         a 
                         137 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         137 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         137 
                       
                     
                   
                   
                     
                       d 
                       137 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 138 connects the right Trapezium to the right Scaphoid. The d 138  length of the right Trapezium runs from the left head of the Scaphoid to the left head of the Trapezium. 
     
       
         
           
             
               A 
               138 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               138 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               138 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               138 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               138 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             138 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           138 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               138 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       
                         a 
                         138 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       
                         a 
                         138 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         138 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         138 
                       
                     
                   
                   
                     
                       d 
                       138 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 139 connects the right Trapezoid to the right Capitate. The d 139  length of the right Trapezoid runs from the left head of the Capitate to the left head of the Trapezoid. 
     
       
         
           
             
               A 
               139 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               139 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               139 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               139 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               139 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             139 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           139 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               139 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       
                         a 
                         139 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       
                         a 
                         139 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         139 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         139 
                       
                     
                   
                   
                     
                       d 
                       139 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 140 connects the right Trapezoid to the right Scaphoid. The d 140  length of the right Trapezoid runs from the left head of the Scaphoid to the left head of the Trapezoid. 
     
       
         
           
             
               A 
               140 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               140 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               140 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               140 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               140 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             140 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           140 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               A 
               140 
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       
                         a 
                         140 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       
                         - 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       
                         a 
                         140 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         θ 
                         140 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                         140 
                       
                     
                   
                   
                     
                       d 
                       140 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Joint 141 connects the right Capitate to the right Hamate. The d 141  length of the right Capitate runs from the left head of the Capitate to the left head of the Hamate. 
     
       
         
           
             
               A 
               141 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             141 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           141 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 141 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 141 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 141 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 141 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       141 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             
                               a 
                               141 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             
                               a 
                               141 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               141 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               141 
                             
                           
                         
                         
                           
                             d 
                             141 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 142 connects the right Hamate to the right Triquetral. The d 142  length of the right Hamate runs from the left head of the Triquetral to the left head of the Hamate. 
     
       
         
           
             
               A 
               142 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             142 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             142 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             142 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             142 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             142 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             142 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             142 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             142 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 142 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 142 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 142 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 142 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       142 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             
                               a 
                               142 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             
                               a 
                               142 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               142 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               142 
                             
                           
                         
                         
                           
                             d 
                             142 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 143 connects the right Triquetral to the right Lunate. The d 143  length of the right Lunate runs from the right head of the Scaphoid to the left head of the Lunate. 
     
       
         
           
             
               A 
               143 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             143 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             143 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             143 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             143 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             143 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             143 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             143 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             143 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                               143 
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 143 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 143 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 143 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       143 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             
                               a 
                               143 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             
                               a 
                               143 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               143 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               143 
                             
                           
                         
                         
                           
                             d 
                             143 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 144 connects the right Lunate to the right Capitate. The d 144  length of the right Lunate runs from the left head of the Lunate to the left head of the Capitate. 
     
       
         
           
             
               A 
               144 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                         
                           
                             - 
                             
                               sin 
                               144 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             144 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           144 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 144 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 144 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 144 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 144 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       144 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             
                               a 
                               144 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             
                               a 
                               144 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               144 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               144 
                             
                           
                         
                         
                           
                             d 
                             144 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 145 connects the right Capitate to the right Scaphoid. The d 145  length of the right Capitate runs from the right head of the Scaphoid to the head of the Scaphoid. 
     
       
         
           
             
               A 
               145 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             145 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             145 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             145 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             145 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             145 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             145 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             145 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             145 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 145 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 145 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 145 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 145 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       145 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             
                               a 
                               145 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             
                               a 
                               145 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               145 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               145 
                             
                           
                         
                         
                           
                             d 
                             145 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 146 connects the right Scaphoid to the right Radius. The d 146  length of the right Scaphoid runs from the left head of the right Radius to the head of the Radius. 
     
       
         
           
             
               A 
               146 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             146 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           146 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 146 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 146 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 146 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 146 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       146 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             
                               a 
                               146 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             
                               a 
                               146 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               146 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               146 
                             
                           
                         
                         
                           
                             d 
                             146 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 147 connects the right Radius to the right Ulna. The d 147  length of the right Radius runs from the left head of the Scaphoid to the left head of the Trapezium. 
     
       
         
           
             
               A 
               147 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               `147 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             147 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           147 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 147 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 147 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 147 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 147 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       147 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             
                               a 
                               147 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             
                               a 
                               147 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               147 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               147 
                             
                           
                         
                         
                           
                             d 
                             147 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 148 connects the right Ulna to the right Lunate. The d 148  length of the right Ulna runs from the left head of the Ulna to the right head of the Lunate. 
     
       
         
           
             
               A 
               148 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             148 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             148 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             148 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             148 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             148 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             148 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             148 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             148 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 148 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 148 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 148 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 148 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       148 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             
                               a 
                               148 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                             148 
                           
                         
                         
                           
                             
                               cos 
                               148 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             
                               a 
                               148 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               148 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               148 
                             
                           
                         
                         
                           
                             d 
                             148 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 149 connects the left thumb Distal Phalanx to the left thumb Proximal Phalanx, forming the 1 st  distal interphalangeal joint (DIP). The d 149  length of the left Distal Phalanx runs through the centroid of the Distal Phalanx from the head of the left Proximal Phalanx to the head of the left distal phalanx 
     
       
         
           
             
               A 
               149 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             d 
                             149 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           149 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 149 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 149 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 149 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 149 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       149 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             
                               a 
                               149 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             
                               a 
                               149 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               149 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               149 
                             
                           
                         
                         
                           
                             d 
                             149 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 150 connects the left index finger Distal Phalanx to the left index finger Middle Phalanx, forming the 2 nd  distal interphalangeal joint (DIP) The d 150  length of the left index finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the left index finger Proximal Phalanx to the head of the left index finger Distal Phalanx. 
     
       
         
           
             
               A 
               150 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             150 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           150 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 150 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 150 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 150 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 150 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       150 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             
                               a 
                               150 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             
                               a 
                               150 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               150 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               150 
                             
                           
                         
                         
                           
                             d 
                             150 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 151 connects the left middle finger Distal Phalanx to the left middle finger Middle Phalanx, forming the 3 rd  distal interphalangeal joint (DIP). The d 151  length of the left middle finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the left index finger Proximal Phalanx to the dead of the left middle finger Distal Phalanx. 
     
       
         
           
             
               A 
               151 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             151 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           151 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 151 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 151 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 151 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 151 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       151 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             
                               a 
                               151 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             
                               a 
                               151 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               151 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               151 
                             
                           
                         
                         
                           
                             d 
                             151 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 152 connects the left ring finger Distal Phalanx to the left ring finger Middle Phalanx, forming the 4 th  distal interphalangeal joint (DIP). The d 152  length of the left ring finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the left ring finger Proximal Phalanx to the head of the left ring finger Distal Phalanx. 
     
       
         
           
             
               A 
               152 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             152 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           152 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 152 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 152 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 152 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 152 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       152 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             
                               a 
                               152 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             
                               a 
                               152 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               152 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               152 
                             
                           
                         
                         
                           
                             d 
                             152 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 153 connects the left little finger Distal Phalanx to the left little finger Middle Phalanx, forming the 5 th  distal interphalangeal joint (DIP). The d 153  length of the left little finger Digital Phalanx runs through the centroid of the Distal Phalanx from the head of the left little finger Proximal Phalanx to the head of the left little finger Distal Phalanx. 
     
       
         
           
             
               A 
               153 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             153 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           153 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 153 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 153 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 153 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 153 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       153 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             
                               a 
                               153 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             
                               a 
                               153 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               153 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               153 
                             
                           
                         
                         
                           
                             d 
                             153 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 154 connects the left index finger Middle Phalanx to the left index finger Proximal Phalanx forming the 2 nd  proximal interphalangeal joint (PIP). The d 154  length of the left index finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the left index finger Proximal Phalanx to the head of the left index finger middle phalanx. 
     
       
         
           
             
               A 
               154 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             154 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           154 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 154 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 154 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 154 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 154 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       154 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             
                               a 
                               154 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             
                               a 
                               154 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               154 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               154 
                             
                           
                         
                         
                           
                             d 
                             154 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 155 connects the left middle finger Middle Phalanx to the left middle finger Proximal Phalanx, forming the 3 rd  proximal interphalangeal joint (PIP). The d 155  length of the left middle finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the left middle finger Proximal Phalanx to the head of the left middle finger Middle Phalanx. 
     
       
         
           
             
               A 
               155 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             155 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           155 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 155 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 155 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 155 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 155 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       155 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             
                               a 
                               155 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             
                               a 
                               155 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               155 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               155 
                             
                           
                         
                         
                           
                             d 
                             155 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 156 connects the left ring finger Middle Phalanx to the left ring finger Proximal Phalanx, forming the 4 th  proximal interphalangeal joint (PIP). The d 156  length of the left ring finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the left ring finger Proximal Phalanx to the head of the left ring finger Middle Phalanx. 
     
       
         
           
             
               A 
               
                 1 
                  
                 5 
                  
                 6 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             
                               1 
                                
                               5 
                                
                               6 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           
                             1 
                              
                             5 
                              
                             6 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   1 
                                    
                                   5 
                                    
                                   6 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   1 
                                    
                                   5 
                                    
                                   6 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   1 
                                    
                                   5 
                                    
                                   6 
                                 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 
                                   1 
                                    
                                   5 
                                    
                                   6 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       
                         1 
                          
                         5 
                          
                         6 
                       
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               156 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               a 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               156 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               156 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               a 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               
                                 1 
                                  
                                 5 
                                  
                                 6 
                               
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               156 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               156 
                             
                           
                         
                         
                           
                             d 
                             
                               1 
                                
                               5 
                                
                               6 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 157 connects the left little finger Middle Phalanx to the left little finger Proximal Phalanx, forming the 5 th  proximal interphalangeal joint (PIP). The d 156  length of the left little finger Middle Phalanx runs through the centroid of the Middle Phalanx from the head of the left little finger Proximal Phalanx to the head of the left little finger Middle Phalanx. 
     
       
         
           
             
               A 
               157 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             157 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           157 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 157 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 157 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 157 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 157 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       157 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             
                               a 
                               157 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             
                               a 
                               157 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               157 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               157 
                             
                           
                         
                         
                           
                             d 
                             157 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 158 connects the left thumb Proximal Phalanx to the left thumb Metacarpal in the 1 st  metacarpophalangeal joint (MCP) The d 158  length of the left thumb Middle Phalanx runs through the centroid of the Middle Proximal Phalanx from the head of the thumb Metacarpal to the head of the thumb Proximal Phalanx. 
     
       
         
           
             
               A 
               158 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             158 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           158 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 158 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 158 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 158 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 158 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       158 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             
                               a 
                               158 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             
                               a 
                               158 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               158 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               158 
                             
                           
                         
                         
                           
                             d 
                             158 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 159 connects the left index finger Proximal Phalanx to the left index finger Metacarpal in the 2 nd  metacarpophalangeal joint (MCP). The d 159  length of the left index finger Proximal Phalanx runs through the centroid of the index finger Proximal Phalanx from the head of the index finger Metacarpus to the head of the index finger Proximal Phalanx. 
     
       
         
           
             
               A 
               159 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             159 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           159 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 159 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 159 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 159 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 159 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       159 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             
                               a 
                               159 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             
                               a 
                               159 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               159 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               159 
                             
                           
                         
                         
                           
                             d 
                             159 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 160 connects the left middle finger Proximal Phalanx to the left middle finger Metacarpal in the 3 rd  metacarpophalangeal joint (MCP). The d 160  length of the left middle finger Proximal Phalanx runs through the centroid of the left middle finger Proximal Phalanx from the head of the middle finger Metacarpus to the head of the middle finger Proximal Phalanx. 
     
       
         
           
             
               A 
               160 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             160 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           160 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 160 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 160 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 160 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 160 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       160 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             
                               a 
                               160 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             
                               a 
                               160 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               160 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               160 
                             
                           
                         
                         
                           
                             d 
                             160 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 161 connects the left ring finger Proximal Phalanx to the left ring finger Metacarpal in the 4 th  metacarpophalangeal joint (MCP). The d 161  length of the left ring finger Proximal Phalanx runs through the centroid of the left ring finger Proximal Phalanx from the head of the ring finger Metacarpus to the head of the ring finger Proximal Phalanx. 
     
       
         
           
             
               A 
               161 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             161 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           161 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 161 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 161 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 161 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 161 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       161 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             
                               a 
                               161 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             
                               a 
                               161 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               161 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               161 
                             
                           
                         
                         
                           
                             d 
                             161 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 162 connects the left little finger Proximal Phalanx to the left little finger Metacarpal in the 5 th  metacarpophalangeal joint (MCP). The d 162  length of the left little finger Proximal Phalanx runs through the centroid of the left little finger Proximal Phalanx from the head of the little finger Metacarpus to the head of the little finger Proximal Phalanx. 
     
       
         
           
             
               A 
               162 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             162 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           162 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 162 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 162 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 162 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 162 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       162 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             
                               a 
                               162 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             
                               a 
                               162 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               162 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               162 
                             
                           
                         
                         
                           
                             d 
                             162 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 163 connects the left thumb Metacarpal to the left Trapezium. The d 163  length of the left thumb Metacarpal runs through the centroid of the left thumb Metacarpal. 
     
       
         
           
             
               A 
               163 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             163 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           163 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 163 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 163 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 163 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 163 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       163 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             
                               a 
                               163 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             
                               a 
                               163 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               163 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               163 
                             
                           
                         
                         
                           
                             d 
                             163 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 164 connects the left thumb Metacarpal to the left Index finger Metacarpal. The d 164  length of the left thumb Metacarpal runs through the centroid of the left thumb Metacarpal, beginning at the head of the left Trapezium, and extending to the head of the left thumb Metacarpal. 
     
       
         
           
             
               A 
               164 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           θ 
                           164 
                         
                       
                     
                     
                       
                         
                           - 
                           sin 
                         
                          
                         
                             
                         
                          
                         
                           θ 
                           164 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           164 
                         
                       
                     
                     
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           θ 
                           164 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
                
               
                   
               
                
               middle 
                
               
                   
               
                
               
                 
                   finger 
                    
                   
                       
                   
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           d 
                           164 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           164 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 164 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 164 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 164 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 164 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       164 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             
                               a 
                               164 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             
                               a 
                               164 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               164 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               164 
                             
                           
                         
                         
                           
                             d 
                             164 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 165 connects the left index finger Metacarpal to the left middle finger Metacarpal. The d 165  length of the left index finger Metacarpal runs through the centroid of the left index finger Metacarpal beginning at the side of the left middle finger Metacarpal. 
     
       
         
           
             
               A 
               165 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             165 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           165 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 165 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 165 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 165 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 165 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       165 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             
                               a 
                               165 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             
                               a 
                               165 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               165 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               165 
                             
                           
                         
                         
                           
                             d 
                             165 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 166 connects the left middle finger Metacarpal to the left ring finger Metacarpal. The d 166  length of the left middle finger Metacarpal runs through the left middle finger Metacarpal beginning at the side of the left ring finger Metacarpal to the head of the left middle finger Metacarpal. 
     
       
         
           
             
               A 
               166 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             166 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           166 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 166 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 166 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 166 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 166 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       166 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             
                               a 
                               166 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             
                               a 
                               166 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               166 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               166 
                             
                           
                         
                         
                           
                             d 
                             166 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 167 connects the left index finger Metacarpal to the Left Trapezium. The d 167  length of the left index finger Metacarpal runs through the left index finger Metacarpal beginning at the head of the left trapezium to the head of the left index finger Metacarpal. 
     
       
         
           
             
               A 
               167 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             167 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           167 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 167 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 167 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 167 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 167 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       167 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             
                               a 
                               167 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             
                               a 
                               167 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               167 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               167 
                             
                           
                         
                         
                           
                             d 
                             167 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 168 connects the left index finger Metacarpal to the left Trapezoid. The d 167  length of the left index finger Metacarpal runs through the left index finger Metacarpal beginning at the head of the left trapezoid to the head of the left index finger Metacarpal. 
     
       
         
           
             
               A 
               168 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             168 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           168 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 168 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 168 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 168 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 168 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       168 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             
                               a 
                               168 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             
                               a 
                               168 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               168 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               168 
                             
                           
                         
                         
                           
                             d 
                             168 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 169 connects the left middle finger to the left Trapezoid. The d 169  length of the left index finger Metacarpal runs through the left index finger Metacarpal beginning at the head of the left trapezoid to the head of the left index finger Metacarpal. 
     
       
         
           
             
               A 
               169 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             169 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           169 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 169 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 169 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 169 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 169 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       169 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             
                               a 
                               169 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             
                               a 
                               169 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               169 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               169 
                             
                           
                         
                         
                           
                             d 
                             169 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 170 connects the left middle finger Metacarpal to the left Capitate. The d 170  length of the left index finger Metacarpal runs through the left index finger Metacarpal beginning at the head of the left Capitate to the head of the left index finger Metacarpal. 
     
       
         
           
             
               A 
               170 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             170 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           170 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 170 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 170 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 170 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 170 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       170 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             
                               a 
                               170 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             
                               a 
                               170 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               170 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               170 
                             
                           
                         
                         
                           
                             d 
                             170 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 171 connects the left middle finger Metacarpal to the left ring finger Metacarpal. The d 171  length of the left middle finger Metacarpal runs through the left middle finger Metacarpal beginning at the head of the left Metacarpal to the head of the left middle finger Metacarpal. 
     
       
         
           
             
               A 
               171 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             171 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           171 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 171 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 171 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 171 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 171 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       171 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             
                               a 
                               171 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             
                               a 
                               171 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               171 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               171 
                             
                           
                         
                         
                           
                             d 
                             171 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 172 connects the left ring finger Metacarpal to the Left Capitate. The d 172  length of the left ring finger Metacarpal runs through the left middle finger Metacarpal beginning at the head of the left Metacarpal to the head of the left Capitate. 
     
       
         
           
             
               A 
               172 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             172 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           172 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 172 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 172 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 172 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 172 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       172 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             
                               a 
                               172 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             
                               a 
                               172 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               172 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               172 
                             
                           
                         
                         
                           
                             d 
                             172 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 173 connects the left ring finger Metacarpal to the left little finger Metacarpal. The d 173  length of the left ring finger Metacarpal runs through the left middle finger Metacarpal beginning at the touchpoint of the left little finger Metacarpal to the touchpoint on the little finger Metacarpal. 
     
       
         
           
             
               A 
               173 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             173 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           173 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 173 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 173 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 173 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 173 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       173 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             
                               a 
                               173 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             
                               a 
                               173 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               173 
                             
                           
                         
                         
                           
                             d 
                             173 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 174 connects the left ring finger Metacarpal to the left Hamate. The d 174  length of the left ring finger Metacarpal runs through the left middle finger Metacarpal beginning at the touchpoint of the left little finger Metacarpal to the touchpoint on the left Hamate. 
     
       
         
           
             
               A 
               174 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             174 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           174 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 174 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 174 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 174 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 174 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       174 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             
                               a 
                               173 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               173 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             
                               a 
                               173 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               174 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               174 
                             
                           
                         
                         
                           
                             d 
                             174 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 175 connects the left little finger Metacarpal to the left Hamate. The d 175  length of the left little finger Metacarpal runs through the left little finger Metacarpal beginning at the touchpoint of the left little finger Metacarpal to the touchpoint on the left Hamate. 
     
       
         
           
             
               A 
               175 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             175 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           175 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 175 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 175 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 175 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 175 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       175 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             
                               a 
                               175 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             
                               a 
                               175 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               175 
                             
                           
                         
                         
                           
                             d 
                             175 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 176 connects the left Trapezium to the left Trapezoid. The d 176  length of the left Trapezium runs through the left Trapezium beginning at the touchpoint of the left Trapezoid to the touchpoint on the left Trapezium. 
     
       
         
           
             
               A 
               176 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             176 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           176 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 176 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 176 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 176 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 176 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       176 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             
                               a 
                               176 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             
                               a 
                               176 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               176 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               176 
                             
                           
                         
                         
                           
                             d 
                             176 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 177 connects the left Trapezium to the left Scaphoid. The d 177  length of the left Trapezium runs from the left head of the Scaphoid to the left head of the Trapezium. 
     
       
         
           
             
               A 
               177 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             177 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           177 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 177 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 177 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 177 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 177 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       177 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             
                               a 
                               177 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             
                               a 
                               177 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               177 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               177 
                             
                           
                         
                         
                           
                             d 
                             177 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 178 connects the left Trapezoid to the left Capitate. The d 178  length of the left Trapezoid runs from the left head of the Capitate to the left head of the Trapezoid. 
     
       
         
           
             
               A 
               178 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             178 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           178 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 178 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 178 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 178 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 178 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       178 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             
                               a 
                               178 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               175 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             
                               a 
                               178 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               178 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               178 
                             
                           
                         
                         
                           
                             d 
                             178 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 179 connects the left Trapezoid to the left Scaphoid. The d 179  length of the left Trapezoid runs from the left head of the Scaphoid to the left head of the Trapezoid. 
     
       
         
           
             
               A 
               179 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             179 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           179 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 179 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 179 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 179 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 179 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       179 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             
                               a 
                               179 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             
                               a 
                               179 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               179 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               179 
                             
                           
                         
                         
                           
                             d 
                             179 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 180 connects the left Capitate to the left Hamate. The d 180  length of the left Capitate runs from the left head of the Capitate to the left head of the Hamate. 
     
       
         
           
             
               A 
               180 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             180 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           180 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 180 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 180 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 180 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 180 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       180 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             
                               a 
                               180 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             
                               a 
                               180 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               180 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               180 
                             
                           
                         
                         
                           
                             d 
                             180 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 181 connects the left Hamate to the left Triquetral. The d 181  length of the left Hamate runs from the left head of the Triquetral to the left head of the Hamate. 
     
       
         
           
             
               A 
               181 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             181 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             181 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             181 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             181 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             181 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             181 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             181 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             181 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ψ 
                               181 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               181 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               181 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               181 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
 
                     
                      
                     
                       
                         A 
                         181 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   181 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   181 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   181 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   181 
                                 
                               
                             
                             
                               
                                 d 
                                 181 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     Joint 182 connects the left Triquetral to the left Lunate. The d 182  length of the left Lunate runs from the left head of the Scaphoid to the left head of the Lunate. 
     
       
         
           
             
               A 
               182 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             182 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             182 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             182 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             182 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             182 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             182 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             182 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             182 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ψ 
                               182 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               182 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               182 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               182 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
 
                     
                      
                     
                       
                         A 
                         182 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   182 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   182 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   182 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   182 
                                 
                               
                             
                             
                               
                                 d 
                                 182 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     Joint 183 connects the left Lunate to the left Capitate. The d 183  length of the left Lunate runs from the left head of the Lunate to the left head of the Capitate. 
     
       
         
           
             
               A 
               183 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             183 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           183 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 183 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 183 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 183 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 183 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       183 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             
                               a 
                               183 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             
                               a 
                               183 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               183 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               183 
                             
                           
                         
                         
                           
                             d 
                             183 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 184 connects the left Capitate to the left Scaphoid. The d 184  length of the left Capitate runs from the left head of the Scaphoid to the head of the Scaphoid. 
     
       
         
           
             
               A 
               184 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             184 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             184 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             184 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             184 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             184 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             184 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             184 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             184 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ψ 
                               184 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               184 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               184 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               184 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
 
                     
                      
                     
                       
                         A 
                         184 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   184 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   184 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   184 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   184 
                                 
                               
                             
                             
                               
                                 d 
                                 184 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     Joint 185 connects the left Scaphoid to the left Radius. The d 185  length of the left Scaphoid runs from the left head of the left Radius to the head of the Radius. 
     
       
         
           
             
               A 
               185 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                         
                           
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                              
                             
                                 
                             
                              
                             
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                               185 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             185 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           185 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 185 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 185 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 185 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 185 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       185 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                              
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                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             
                               a 
                               185 
                             
                              
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                              
                             
                                 
                             
                              
                             
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                               185 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                              
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                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             
                               a 
                               185 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               185 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               185 
                             
                           
                         
                         
                           
                             d 
                             185 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 186 connects the left Radius to the left Ulna. The d 186  length of the left Radius runs from the left head of the Scaphoid to the left head of the Trapezium. 
     
       
         
           
             
               A 
               186 
             
             = 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           
                             d 
                             186 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           a 
                           186 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                   
                               
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                                 α 
                                 186 
                               
                             
                           
                           
                             
                               
                                 - 
                                 sin 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 186 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 186 
                               
                             
                           
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 186 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       
 
                     
                      
                     
                       A 
                       186 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             
                               a 
                               186 
                             
                              
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                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                       
                       
                         
                           
                             sin 
                              
                             
                                 
                             
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                               θ 
                               186 
                             
                           
                         
                         
                           
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                              
                             
                                 
                             
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                               186 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             
                               a 
                               186 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               186 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               186 
                             
                           
                         
                         
                           
                             d 
                             186 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Joint 187 connects the left Ulna to the left Lunate. The d 187  length of the left Ulna runs from the left head of the Ulna to the left head of the Lunate. 
     
       
         
           
             
               A 
               187 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             187 
                           
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             ϕ 
                             187 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             187 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             187 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             187 
                           
                         
                       
                       
                         0 
                       
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             187 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                            
                           
                               
                           
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                             θ 
                             187 
                           
                         
                       
                       
                         0 
                       
                       
                         
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                            
                           
                               
                           
                            
                           
                             θ 
                             187 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               ψ 
                               187 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               187 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ϕ 
                               187 
                             
                           
                         
                         
                           
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                              
                             
                                 
                             
                              
                             
                               ϕ 
                               187 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
 
                     
                      
                     
                       
                         A 
                         187 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   187 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 
                                   a 
                                   187 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   187 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   187 
                                 
                               
                             
                             
                               
                                 d 
                                 187 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     The above matrices and the model that uses these can be stored in one or more databases of the system and can be used in the determination of a patient&#39;s condition or the progression of any conditions the patient may have. In addition, the above matrices and model can be used to determine, in conjunction with data from the foot based data gathering device(s), skeletal joint angles, force vectors, and the propagation of such force vectors through the various components of the skeletal/joint system. As well, the above model and matrices can be used to determine forces at each of the various joints and to interpret these forces into joint motions that indicate normalization of propagated force vectors through the kinematic chain. These can then be used to update/create/adjust a kinematic model specific to each individual/patient. 
     The system of the present invention can also use the above matrices and model to create and update a fully populated kinematic model for a specific user/patient. This can be done when the user/patient initially uses the system and baseline data for that user/patient is gathered. Information obtained during registration/initial use of the system may include the sensor data from the data collection sensors obtained when the user first activates the sensor package. As should be clear, to gather baseline/initial data for a user, the user takes a fixed number of steps and the data gathered for these steps can form the basis for a baseline data set or biometric loop signature. 
     The system can also include one or more mechanisms for ingesting pressure map data, pressure loop data, thermal map data, thermal loop data, GPS receiver data, and 9-degree of freedom Inertial Navigation System data. Such data can then be used with a pose estimation software module that may be internal or external to the system of the present invention. 
     Once the system has the baseline data and one or more data sets obtained subsequent to the gathering of the baseline data, the one or more data sets can be compared to the baseline data. This may involve comparing the original registration kinematic model (the base kinematic chain model for a specific individual) to a currently obtained kinematic model (for the same individual) to determine gait differences between the original baseline data and any subsequent data sets. Gait differences may include kinematic joint specific changes in rotation. These gait differences may be used to update the individual&#39;s kinematic model based on the gait-based sensor data interpreted as force vectors through the kinematic model. 
     It should be clear that loop data is reflective of all gait based biometric contributors including foot shape characteristics, foot type characteristic, foot placement characteristic, weight and load bearing weight, step count, step length, stride length, stride to stride variability, cadence, foot to floor forces, center of force, foot distance apart (instantaneous, average and standard deviation), plantar pressure distribution (instantaneous, average and standard deviation), and plantar peak pressure distribution (instantaneous, average and standard deviation). Loop data characteristics of the biometric signature includes hidden contributors such as those described in the kinematic chain model and any change in the kinematic chain (such as, for example, an ankle injury, toe, knee or hip injury or impairment from disease such as arthritis) will adversely affect the loop data formations of the biometric signature. Such adverse effects on the loop data formations will therefore signal an abnormality indicative of a regression or progression of a disease, injury, or sickness or the onset of a new condition. As noted in this disclosure, such disease, injury, sickness, or new condition would be provisioned by way of a distributed data analysis system using machine learning and artificial intelligence methods and techniques. An example of one human stride represents the correlation of loop biometric signature data (the resulting biometric loop signature data following the steps illustrated in  FIGS. 3, 4, 5, 6, 7, and 8 ), and the kinematic chain model is described below:
         (i) Movement during swing phase—the shoulder extends, the spine rotates right, the pelvis rotates left (passive), the hip flexes, the knee flexes, then extends, the ankle dorsiflexes, the foot supination (inversion) and the toes extend.   (ii) Static Position at initial swing—the shoulder is flexed, the spine is rotated left, the pelvis is rotated right, the hip is slightly extended and internally rotated, the knee is slightly flexed, the ankle is fully plantarflexed, the foot is supinated, and the toes are slightly flexed.   (iii) Static Position at Midswing—the shoulder is neutral, the spine is neutral, the pelvis is neutral, the hip is neutral, the knee is flexed 60-90°, the ankle is plantar flexed to neutral, the foot is neutral, and the toes are slightly extended.   (iv) Static Position at Terminal Swing—the shoulder is extended, the spine is rotated right, the pelvis is rotated left, the hip is flexed and externally rotated, the knee is fully extended, the ankle is fully dorsiflexed, the foot is neutral, and the toes are slightly extended.   (v) Static Position at Toe-Off—the shoulder is flexed, the pelvis is rotated right, the hip is fully extended and internally rotated, the knee is fully extended, the ankle is plantarflexed, the foot is fully supinated, and the toes are fully extended.       

     As noted in this disclosure, a gait-based data analysis system that includes the kinematic chain model explained and noted above allows for a continued and fulsome analysis of human movement and mobility. In addition, it also allows for a determination and monitoring of physiological and behavioral contributors to normal gait and to gait abnormalities, thereby leading to highly individualized gait-based biometric analysis. 
     The system may also provide some user interface enhancements to allow medical/health personnel to view any changes in the patient&#39;s gait/data. As an example, the user interface enhancements may include a color coding based mechanism that allows the medical/health personnel to observe/see a dynamic view of the original registration kinematic model (i.e., the base kinematic chain model) in conjunction with a current version of the individual&#39;s kinematic model (i.e., a kinematic chain model derived from a later received data set). The color coding based mechanism can then be used to highlight/identify differences between the two models. 
     The system may also include other user interface enhancements that allow health/medical personnel to view, dynamically, a current kinematic model for an individual with one or more gait-base kinematic models for specific gait types. These gait types may include: nominal gait models, pain models, injury models, surgery models, weakness models, balance deficit models, pronation models, supination models, over pronation models, and over supination models. 
     In addition to the above, the system may be used to assess a user/patient&#39;s gait pattern based on the current kinematic model. The gait pattern may be determined to be one of: Normal Gait Patterns, Spastic Paraparetic Gait Patterns, Cerebellar ataxias Gait Patterns, Parkinsonian Gait Patterns, Frontal Gait Patterns, Antalgic Gait Patterns, Trendelenburg Gait Patterns, Rheumatoid Gait Patterns, Sensory Gait Patterns, Hemiplegic Gait Patterns, Diplegic Gait Patterns, Myopathic Gait Patterns, Neuropathic Gait Patterns, Flexed Knee Gait Patterns, Cautious Gait Patterns, Hypotonic Gait Patterns, Spastic Gait Patterns, Dyskinetic Gait Disorder Patterns, Dementia Gait Patterns, Diabetic Gait Patterns, Flexed Knee Gait Patterns, Cautious Gait Patterns, Hypotonic Gait Patterns, Spastic Gait Patterns, Vestibular Gait Patterns, and Alzheimer&#39;s Gait Patterns. 
     It should be clear that the systems and methods of the present invention may use machine learning techniques and artificial intelligence techniques and subsystems to perform continuous analysis of kinematic contributors to a user/patient&#39;s gait. These kinematic contributors may include all 26 bones and 31 joints in each foot and ankle, up through the lower leg, upper leg, pelvis, sacral vertebrae, and lumbar vertebrae. 
     The user/patient&#39;s gait and the correlation of these contributors to multiple conditions, environments, and surrounding circumstances can be determined using the present invention&#39;s systems and methods. These conditions, environments, and surrounding circumstances may include disease, injury and sickness, rehabilitation, therapeutics, and effects of prescribed pharmacologies. 
     Regarding the programming or storage of the signature or characteristic data into the system, this is preferably done when the user first registers and wears the insole component of the system. This first data set can provide a baseline set of data to be used in comparison with subsequent data sets. This is done by having the user use the insole/sensor module by taking a specific number of normal steps. These steps are then captured in the system and are stored as signature/characteristic or baseline data. Once stored, the signature data can be retrieved and various characteristics of the signature data (by way of the signature loop) can be determined as described above. As described above, the signature data stored may take any number of forms. The signature data may be the raw data gathered from the user when s/he took the specific number of normal steps. Alternatively, the signature data may be the filtered version of the raw data or it may be the various characteristics of the various possible signature loops. Also, instead of the raw data which forms the waveforms, the waveforms themselves may be stored as signature data. The signature data may take any form as long as the characteristics of the signature loops may be derived from or be extracted from the signature/characteristic/baseline data. 
     As noted above, the system may include a server and a network of distributed databases connected to that server. The data gathered by the insole may be transmitted to either the server or to the databases. Such a server or its connected network of distributed databases may, to assist in the diagnosis of the user&#39;s condition for example, be used to consult with at least one medical or fitness database. Referring to  FIG. 9 , a block diagram illustrating the connections between the various parts of the system is illustrated. As can be seen from  FIG. 9 , the user&#39;s insole data gathering sub-system  20  communicates with a server  50 . The server  50  serves as the main data processing and analytical unit to determine for example the user&#39;s physical or medical condition based, at least in part, on the data gathered from the user&#39;s gait. After the server receives gait data or the characteristic data from the insole  20  through communications module  50  (perhaps from two insoles), this data can be used by the server to determine the user&#39;s physical or medical condition. This includes determining the condition&#39;s progression, regression, or development. This may be performed by referring to one or more medical, fitness or kinematic reference model databases  60 A,  60 B,  60 C,  60 D. The server may receive both structured and unstructured data from the databases and/or from the user&#39;s insole. Preferably, the databases contain data relating to disease pathology, human performance, workforce industrial safety, human skeleton kinematics, sports performance, post-operative rehabilitation, therapeutics, pharmaceuticals prescribe and known side effects of such prescribed pharmaceuticals and/or gait-based biometric data relating to injury and disease pathologies including conditions such as diabetes, Parkinson&#39;s disease, dementia, aging, and other vestibular disorders. 
     To assist the server in determining a diagnosis and/or a determination of the progression, regression, or change in the user&#39;s condition, the following characteristics of the user may be entered into the server and may be taken into account in any analysis: a foot type, general health, fitness level, type of gait, height, weight, age, gender, and/or nationality. Data entry into the server may be performed using various well-known means. As an example, such data may be transferred from a user&#39;s profile on a connected mobile device to the server. The server may then take such user data, in conjunction with the gait-based and gait-derived data and analyze such data with data from the various databases. Then, based on the input from the various databases and the gathered data for the user, the server can produce its output. 
     The server&#39;s output may include an indication that the user&#39;s condition has regressed, progressed, is abnormal, or is normal. Similarly, the output may indicate the pathologies operative with the user as well as an indication if the user requires or is using corrective orthotics. 
     It should be noted that the user&#39;s foot characteristic may be one of: Egyptian, Roman, Greek, Germanic, or Celtic. Similarly, the user&#39;s foot type may be one of: flat arch, medium arch, or high arch. The user&#39;s type of gait may be one of: normal, toe in or toe out, knee in or knee out, lean forward or lean backward, posture easy or posture rigid, and trunk sway. The user&#39;s health and/or fitness may be categorized as one of: athletic, fit, average, below average, poor, or one where the user has a reported illness or disease. 
     Referring to  FIG. 10 , a flowchart of one aspect of the present invention is illustrated. The process illustrated in  FIG. 10  details the steps in one variant of an aspect of the present invention. To store the signature data into the system, the initial step  100  in the process is that of selecting two of the sensors to be used in the comparison process. As noted above, the sensors are, in one embodiment, inserted or installed in a user&#39;s shoe. Once the sensors have been selected, data is gathered from these sensors as the user walks normally (step  110 ). Once gathered from the sensors, the data is then correlated with one another to form the data pairs noted above (step  120 ). This means that data points from one sensor is mated with data points from another sensor. With the data pairs in hand, at least one characteristic loop can then be created/derived from the data pairs (step  130 ). Depending on the configuration, discrete steps may be separated from one another so that each step may have its own characteristic loop. Alternatively, an average characteristic loop may be derived from the data values from the sensors. Once the average characteristic loop has been found (step  140 ), the signature data can be retrieved (step  150 ). The signature data, depending on configuration can then be used to determine the signature loop (step  160 ). The characteristics of both the average characteristic loop and the signature loop can then be calculated or derived from the two sets of data (step  170 ). The characteristics are then compared (step  180 ), taking into consideration the preprogrammed tolerances. If the characteristics from the two sets of data are the same (step  190 ) (within the preprogrammed tolerances) then a match is found (step  200 ) indicating no change in the user&#39;s condition (step  210 ). If they are not within the preprogrammed tolerances, then no match is found (step  220 ) and a potential change in the user&#39;s condition is indicated (step  230 ). The indication of whether a match is found or not found can then be communicated with the server  50 . The server  50  may then communicate with the distributed databases to gather data and/or correlations between the user&#39;s characteristic data and potential/actual medical and/or physical conditions. 
     In another variant, the process according to one aspect of the invention may be seen as eight specific steps. These steps are described below. 
     The first processing step after retrieving the data is one where the pair sensor signals are filtered applying DFT (Discrete Fourier Transform) based low-pass filter. The cut-off frequency of the filter is defined taking into account a Nyquist frequency (related to the sampling rate) on the high end, and a main signal frequency (related to the walking speed of the individual) on the low end. Walking frequency estimation is also a part of the described processing step. 
     Using an FFT (Fast Fourier Transform) implementation technique and sync-filter as a benchmark, a low pass filter with flat pass-band (low ripple) high stop band attenuation may be used. Additional advantage is taken from the use of non-causal filters since the hard-real-time processing is not required (signals are registered first and then filters are applied). 
     The second processing step is a construction of the characteristic loop for the chosen pair of signals. The characteristic loop is an ordered set of points with coordinates (X(i),Y(i)) where X(i) is a first chosen signal and Y(i) is a second chosen signal, i is an index corresponding to the sample number. 
     An autonomous loop is constructed for the time period (subset of all samples) corresponding to the evolution of both signals from low level to maturity level and back to low level. Such a construction is possible since the low level of all signals have a non-empty intersection corresponding to the foot not contacting the ground. 
     Due to quasi-periodicity of all signals resulting from the nature of human walking, characteristic loops can be constructed autonomously for several periods in time. 
     Although initially defined for raw signals, autonomous loops can then be constructed for smoothed signals (obtained after the first step processing described above). 
     The third processing step is that of averaging the loops. Several loops are constructed according to the recording of several steps while the person is walking. Those steps and respectively those loops are subject to significant variations. It has been found that only the average loop provides a stable and robust characteristic of human walking. 
     Averaging of the loops is done by artificially synchronizing several loops (as corresponding to several steps) followed by weighted averaging of the synchronized loops. Weight factors are computed according to the phase shifts from an estimated reference signal (main walking frequency—as per first processing step). 
     The fourth processing step consists of extracting initial geometrical parameters from the average loop such as loop length, loop width, direction of longitudinal axes, loop directionality (clockwise or counterclockwise) and the area inside the loop. Other characteristics/parameters which can be used are the variance of each parameter listed above as computed for individual walking steps and as compared to the average value (computed from average loop). 
     Other parameters which can be extracted may use:
         Geometrical method—identify a point on the loop farthest from the origin (e.g. referred to as M in this example) this point is further used to find the length (IOMI) and direction of the longitudinal axis (OM), the width is defined as maximal projection onto the line perpendicular to OM   Statistical approach—considering the loop as the cloud of points, the elliptical fit (correlation analysis) can be applied followed by extraction of the parameters of the fitted ellipse (major and minor axis length and orientation).       

     Regarding loop directionality, the directionality of the loop is related to the phase shift between signal Y and signal X. Namely, the loop is clockwise if Y signal grows from low level to maturity first, followed by the growth of X signal. 
     The fifth processing step consists of analysing special cases. It is worth noticing that in some cases, for some pairs of signals, the construction of the loop as described above might yield less than perfect results. This may result in a “degenerated loop” due to a high correlation between signals. The “loop” in such case is located very close to the diagonal. For this case only the point farthest from the origin is actually computed (corresponding to maximal amplitude of both signals). 
     The sixth processing step consists of comparing the loops computed from two separately recorded data sets. It has been found that the parametric representation of the pair-wise average loops have a high discrimination efficiency (see  FIG. 8  as an example). Namely, for several pairs of signals/sensors extracted from the set of 8 signals/sensors, the average loops constructed from the smoothed signals stably demonstrate significant similarities when constructed from the data corresponding to the same individual as well as significant differences from average loops constructed for different individuals. 
     The seventh processing step consists of combining the results of the comparison of several (up to all 56 possible pairs from  8  different sensors/signals) pairs in order to produce a highly efficient discriminate function. Results from various pairs are first weighted according to the number of parameters that can be robustly estimated to support the comparison of the loops. Finally, the results from various pairs can be fused using a Dempster-Shaefer framework for an estimation of the likelihood that loops from the baseline data and the gathered data are similar or not. 
     In addition to the above processing steps, it should be noted that, for mobility-impaired-based applications, the data gathered can be expected to have a number of behaviours. The characteristics that are extracted when performing the loop signature computation (see above) can be divided in three classes: 
     A) (Class-1) Dimensionless parameters such as:
         (1) loop directionality,   (2) direction of longitudinal axes of the loop,   (3) loop elongation (e.g. major to minor axis ratio), etc., as well as standard deviations of those parameters computed over all of the collected data.       

     B) (Class-2) Size-type parameters having a single dimension such as:
         (1) loop length,   (2) loop width,   (3,4) major and minor axis of elliptical approximation of the loop (see above), etc. as well as standard deviations of those parameters computed over all of the collected data.       

     C) (Class-3) Area-type parameters having two dimensions such as:
         (1) area of the loop,   (2) product of major and minor axis of elliptical approximation of the loop and variance of those parameters computed over all of the collected steps.       

     These 3 classes of the parameters can be used in different ways in determining the estimation of differences between data sets. For processing the data sets based on mobility impairments, factors such as stride length, stride to stride variability, cadence, and other movement data can affect class 1 and 2 data sets while factors such as weight, posture and other similar physiological changes can affect class 3 parameters. 
     As an example of how physical changes can affect the characteristics of the derived loops, one can look at the effects of weight-based differences. For such differences, dimensionless parameters (Class-1) are expected to be invariant to weight changes. Size-type parameters (Class-2) are expected to be proportional to the weight change, reflecting the fact that the loop is stretched or contracted according to the weight change factor (i.e. the ratio of newly estimated weight to the older one). Area-type parameters (Class-3) are expected to be proportional to the square of the weight change factor. 
     For processing of weight change-based data, the processing steps can be summarized as:
         1. Extraction of the data pairs that provide the robust estimation of relevant parameters;   2. Estimation of the weight change factor from the class-2 (direct) and class-3 (as square root) parameters and verification of invariance of class-1 parameters;   3. Determination of the hypothetical (average) value of the weight or physical characteristic factor;   4. Analysis of the result based on Dempster-Shaefer framework in order to estimate the likelihood that the gathered data supports the determined value.       

     Of course, the above steps can also be used to process data to determine changes in gait-based data due to other physiological changes. In step 2, instead of estimating the weight change factor, the change factor due to the physiological change can be performed and verification that some other parameters are invariant can be performed. 
     It should also be noted that a data histogram of daily loop signatures can be stored in the storage module and can be periodically re-correlated to form a new biometric loop signature which reflects the user&#39;s weight gain or loss or the progression of regression of a disease, sickness, or injury. 
     To assist in the understanding of the present invention,  FIG. 11  is provided. In  FIG. 11  is illustrated the baseline or signature loops for a left foot (left image) and for a right foot (right image) for a user. 
     The system described above may be used in any number of ways. The system can be used to determine a user&#39;s physical or medical condition as well as whether a fitness program or treatment regimen is effective or not. As is well known, for some physical and medical conditions, the progression of the condition affects a person&#39;s gait. Similarly, the regression of the condition also affects the person&#39;s gait. As such, by comparing a user&#39;s baseline gait data with subsequently gathered gait data, the user&#39;s condition can be monitored. If no change in the user&#39;s gait is detected, then the physical or medical condition has neither progressed nor regressed. If there is a noticeable change in the user&#39;s gait (as evidenced by differences in the loops derived from the baseline gait data and the subsequent gait data) this may indicate progression, regression, efficacy of a fitness program or treatment regimen, or any number of health changes in the user. This determination can additionally be performed by the server, perhaps in conjunction with input from the various databases noted above. As well, the determination may be performed after human verification and checking of the data provided. This determination may be made in conjunction with other clinical tests so as to determine correlation between the loop differences, the types of differences, the amount of the difference, and the different conditions and changes in the condition. 
     In another embodiment, all of the data processed by the data processing module may be internally encrypted so that external systems would not be privy to the raw data transferred between the sensor module and the data processing module. Prior to transmitting the raw data from the sensor module to the data processing module, the data may be automatically encrypted. As can be understood, the data processing module may be physically remote from the sensor module and, as such, the data transmissions between these modules may be vulnerable to the outside. In another embodiment, the data processing module is contained within the insole to ensure that any data transfers between the modules are slightly more secure. 
     In another embodiment, any data transfers or communications between the system and any outside server or network systems are encrypted, preferably with one-time encryption schemes, to ensure that outsiders are not able to intercept any usable data. Such precautions would preserve the system user&#39;s privacy. 
     The system of the invention may be used to periodically determine if a user&#39;s physical or medical condition is progressing or regressing. As well, it may be used to determine if any fitness program or treatment regimen to which the user is being subjected to has had an effect on the user or on the user&#39;s gait. The user&#39;s baseline gait data may be gathered when the user first visits a facility properly equipped with the system of the invention. Subsequent visits by the user would entail gathering subsequent gait data sets. The loops derived from the baseline gait data set and the subsequent gait data sets can be compared with one another to view any variances between the user&#39;s gait data. The amount of change in the loop characteristics from the different data sets can provide an indication as to the degree of change in the user&#39;s condition. Large changes in the loop characteristics may indicate an acceleration in the user&#39;s condition and may also indicate whether the user&#39;s fitness program or treatment regimen (which may include pharmacological treatments) is effective or not. 
     It should be clear that the present invention may be used in multiple ways and may be used in different variants. In one variant, the data from the sensors are transmitted to an external device such as a smartphone, smart watch, or home-based computer system for further analysis. A further variant uses the force applied to different areas of the sensor module. The forces are used in the skeletal and joint models and are correlated with a range of disease related gait patterns using models and data stored in the distributed databases. 
     To assist in mapping the existing and possible biomarkers in the gait-based data stored in the distributed databases, the system may use machine learning and artificial intelligence algorithms to continuously search for existing and potential biomarkers that relate to disease pathologies, sicknesses, and injuries, as well as effects of prescribed pharmacologies. 
     It should also be clear that the system may use distributed databases that may be populated using patient data from medicine dispensing organizations (e.g. pharmacies), medical facilities (e.g., hospitals), and/or regulators. Regulators such as the USFDA may provide data as that data relates to published side effects of prescribed pharmaceuticals on the market and available to public consumers. The populated databases can then be mined for conclusions, trends, and insights into the health and wellness of the population using machine learning and artificial intelligence algorithms. These mined conclusions, trends and insights can then be applied to individual and population-based risk reduction models relating to health, medicine, product liability, online behaviors, as well as for insurance purposes. For such insurance uses, the data and the conclusions from the data can provide structured and verifiable data for personalized actuarial sciences and continuous underwriting of risk. As a further use for the system, portions of the system can form the basis of a gait-based identification system. This identification system can then be used to remove or reduce source data fraud. The identification system may be based on using a baseline gait based dataset for a user that is stored in a user device or on a server. When the user approaches a gateway or device where an identity needs to be verified (e.g. an ATM, a secure doorway/location, etc.), the baseline gait based dataset on the user device or server can be checked against the real-time or near real-time data produced by the gait based data gathering apparatus. If the data gathered is not within acceptable parameters, then the person approaching the gateway or device is not authenticated. 
     It should be clear that one or more of the databases of the present invention may be populated with pharmaceutical data from published records of one or more federal, state, provincial, or foreign regulator. This would enable the data processing module of the system to correlate biomarkers with known side effects of medications. These medications may, of course, be for existing conditions of users. 
     In addition to the above, one or more of the databases may be populated with kinematic data from representative mathematical models. This allows the data processing module to correlate the biomarkers with skeletal or joint abnormalities for the users&#39; existing conditions. 
     Yet a further use for the present system is that of determining individualized and population insights derived from the distributed databases. These insights may be developed using machine learning and artificial intelligence algorithms and may be derived using patient data in the distributed databases, with the patient data being sourced from medical and pharmaceutical organizations. 
     It should also be clear that the skeletal and joint models used in the present invention may be used, in conjunction with machine learning and artificial intelligence algorithms, with assistive devices (such as exoskeletons) and with bipedal robotic systems to thereby provide for natural human movement, balance, and adaptive mobility. 
     The system and methods of the present invention may also be used determining athletic performance or in the gathering of athletic performance metrics. The data gathered while an athlete is training may be compared against a baseline dataset gathered prior to the commencement of training. Such comparisons between the baseline dataset or between different datasets taken during training can provide metrics of how an athlete&#39;s performance is progressing or regressing. Of course, other datasets (whether gathered in real-time or near real-time in training or during competitions) can also provide indications as to an athlete&#39;s performance. It should also be clear that the comparisons and the analysis may be performed in real-time or in near real-time depending on the configuration of the data processing system configured to receive and/or process the datasets. Depending on the configuration, real-time performance metrics or near real-time performance metrics may be obtained. 
     A further aspect of the present invention involves the format standardization of loop data characteristics such that the data may be used with the full range of foot size, shape and characteristic including age, gender and nationality for continued analysis and for individual and population generated insights. 
     It should be clear that the various aspects of the present invention may be implemented as software modules in an overall software system. As such, the present invention may thus take the form of computer executable instructions that, when executed, implements various software modules with predefined functions. 
     The embodiments of the invention may be executed by a computer processor or similar device programmed in the manner of method steps or may be executed by an electronic system which is provided with means for executing these steps. Similarly, an electronic memory means such as computer diskettes, CD-ROMs, Random Access Memory (RAM), Read Only Memory (ROM) or similar computer software storage media known in the art, may be programmed to execute such method steps. As well, electronic signals representing these method steps may also be transmitted via a communication network. 
     Embodiments of the invention may be implemented in any conventional computer programming language. For example, preferred embodiments may be implemented in a procedural programming language (e.g., “C” or “Go”) or an object-oriented language (e.g., “C++”, “java”, “PHP”, “PYTHON” or “C #”). Alternative embodiments of the invention may be implemented as pre-programmed hardware elements, other related components, or as a combination of hardware and software components. 
     Embodiments can be implemented as a computer program product for use with a computer system. Such implementations may include a series of computer instructions fixed either on a tangible medium, such as a computer readable medium (e.g., a diskette, CD-ROM, ROM, or fixed disk) or transmittable to a computer system, via a modem or other interface device, such as a communications adapter connected to a network over a medium. The medium may be either a tangible medium (e.g., optical or electrical communications lines) or a medium implemented with wireless techniques (e.g., microwave, infrared or other transmission techniques). The series of computer instructions embodies all or part of the functionality previously described herein. Those skilled in the art should appreciate that such computer instructions can be written in a number of programming languages for use with many computer architectures or operating systems. Furthermore, such instructions may be stored in any memory device, such as semiconductor, magnetic, optical or other memory devices, and may be transmitted using any communications technology, such as optical, infrared, microwave, or other transmission technologies. It is expected that such a computer program product may be distributed as a removable medium with accompanying printed or electronic documentation (e.g., shrink-wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server over a network (e.g., the Internet or World Wide Web). Of course, some embodiments of the invention may be implemented as a combination of both software (e.g., a computer program product) and hardware. Still other embodiments of the invention may be implemented as entirely hardware, or entirely software (e.g., a computer program product). 
     A person understanding this invention may now conceive of alternative structures and embodiments or variations of the above all of which are intended to fall within the scope of the invention as defined in the claims that follow.