Abstract:
A method for estimating joint load at a joint of a segment. The method comprises the steps of receiving kinematic data, determining a modified acceleration using at least the kinematic data, estimating a joint load using at least the modified acceleration; and determining simulated kinematic data for the segment using at least the joint load. The method addresses the problems with conventional inverse dynamics analysis by providing a forward dynamics solution for estimation of joint loads that is stable, guaranteed to converge, computationally efficient, and does not require acceleration computations. According to one embodiment, a joint load is estimated recursively.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 10/151,647 filed on May 16, 2002 now U.S. Pat. No. 7,135,003 which claims priority under 35 U.S.C §119(e) from U.S. provisional applications No. 60/301,891 filed on Jun. 29, 2001 and No. 60/353,378 filed on Jan. 31, 2002 which are all incorporated by reference herein in their entirety. This application is related to U.S. patent application Ser. No. 11/038,978, filed on Jan. 19, 2005, entitled “A System and Method of Estimating Joint Loads Using an Approach of Closed Form Dynamics” which is incorporated by reference herein in its entirety. This application is also related to U.S. patent application Ser. No. 11/038,691, now U.S. Pat. No. 7,469,166 filed on Jan. 19, 2005, entitled “A System and Method of Predicting Novel Motion in a Serial Chain System” which is incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to human motion analysis and synthesis. The present invention relates to inverse dynamics analysis, which uses knowledge of the motion of a dynamical system to determine applied forces and moments at joints. The present invention further relates to forward dynamics analysis, which provides the motion of a biomechanical system as a consequence of applied forces. The present invention further relates to a control theoretic framework for analysis and synthesis of human motion, whereby the estimation of internal forces and moments has been formulated as a trajectory tracking control problem. 
     BACKGOUND OF THE INVENTION 
     Studies of neuromuscular systems are typically concerned with either the synthesis or analysis of human motion (Delp and Loan, A Computational Framework for Simulating and Analyzing Human and Animal Movement, IEEE Computing in Science and Engineering, 2(5): 46-55, 2000; Thelen, Anderson and Delp, Generating Dynamic Simulations of Movement Using Computed Muscle Control, Journal of Biomechanics, 36:321-328, 2003, which are incorporated by reference herein in their entirety). The synthesis problem, referred to as forward dynamics analysis, attempts to provide the motion of a biomechanical system as a consequence of the applied forces and given initial conditions. The analysis, or inverse dynamics problem, can be viewed as the inverse of the synthesis problem and is conventionally used to estimate joint forces and joint moments. One or more forces or moments at a joint are referred to as joint loads. 
     In a conventional inverse dynamics analysis, joint forces and joint moments are calculated from the observation of segmental movement. Inverse dynamics analysis is conventionally applied to biomechanics problems because the internal forces of human joints cannot be readily measured. Segment movements, however, can be measured and joint angles can be inferred from the measured displacement to determine the corresponding joint forces and torques. Thus, inverse dynamics analysis is the conventional method used to gain insight into the net summation of all torques and all muscle activity at each joint. 
     A big challenge with using inverse dynamics in the study of human motion is the error caused by calculating higher-order derivatives to calculate joint forces and moments. Methods for using inverse dynamics concepts in biomechanics are well developed if the input signals are noise-free and the dynamic model is perfect. Experimental observations, however, are imperfect and contaminated by noise. Sources of noise include the measurement device and the joint itself. Inverse dynamics methods for calculating joint forces and moments require the calculation of higher order derivatives of experimental data contaminated by noise, which is a notoriously error prone operation (Cullum, Numerical Differentiation and Regularization, SIAM J. Numer. Anal., 8(2):254-265, 1971, which is incorporated by reference herein in its entirety). Specifically, the angular accelerations for a three-dimensional segment are the second derivatives of its joint angles and the linear accelerations of the segment are the second derivatives of its center of mass coordinates. 
     Numerical differentiation of the experimental observations amplifies the noise. The presence of high frequency noise is of considerable importance when considering the problem of calculating velocities and accelerations. When input signals with noise are differentiated, the amplitude of each of the harmonics increases with its harmonic number. When input signals are differentiated, the velocity signals increase linearly, while accelerations increase in proportion to the square of the harmonic number. For example, second order differentiation of a signal with high frequency noise ω can result in a signal with frequency components of ω 2 . The result of this parabolic noise amplification is erroneous joint force and joint moment calculations. 
     Although numerical schemes are available to provide estimates of higher order derivatives, the reliability of results is limited since there is no optimal solution or automatic method to filter biomechanical data (Giakas and Baltzopoulos, Optimal Digital Filtering Requires a Different Cut-Off Frequency Strategy for the Determination of the Higher Derivatives, Journal of Biomechanics, 30(8):851-855, 1997, which is incorporated by reference herein in its entirety). Although techniques exist for filtering the noise, filtering is difficult and time-consuming because much analysis is required to separate the true signal in the biomechanical data from the noise. For example, low-pass filtering is commonly used to reduce high frequency errors. A difficulty in low-pass filtering, however, is the selection of an optimal cutoff frequency ƒ c . Because there is no general solution for selecting optimal filter parameters, filtering techniques often produce unreliable results. 
     Optimization-based approaches have been proposed to estimate joint forces and joint moments without the errors associated with performing a conventional inverse dynamics analysis (Chao and Rim, Application of Optimization Principles in Determining the Applied Moments in Human Leg Joints during Gait, J. Biomechanics, 6:497-510, 1973, which is incorporated by reference herein in its entirety). Unlike inverse dynamics, optimization-based methods do not require numerical differentiation. However, the application of optimization-based solutions is limited because convergence and stability are not guaranteed, the methods are computationally expensive, and are generally too complex to implement. 
     Another problem with using inverse dynamics for analyzing human motion is that the inverse dynamics technique lacks the capability to predict the behavior of novel motions, a problem typically encountered in clinical applications. In inverse dynamics, forces and moments are calculated from observed responses. The prediction of novel motions involves calculating the response expected from the application of forces and moments. An inverse dynamics analysis lacks predictive capability because forces and moments are calculated rather than the expected response from the application of those forces and moments. 
     Another problem with certain inverse dynamics procedures that utilize only kinematic data is that they utilize whole-body solution. Parametric uncertainties in an upper body portion, including the physical parameters of the upper body portion or the effects of external loads, are significant sources of error in the estimation of joint forces and moments when using closed form, whole body dynamic procedures. 
     For a three-dimensional body, there is therefore a great need for a system and method for estimating joint loads without the errors caused by calculation of higher order derivatives of kinematic data with noise. What is further needed is a system and method for estimating joint forces and moments that does not necessarily require closed form, whole body analysis. Further, there is great need for a system and method for predicting human motions as a consequence of applied forces. 
     SUMMARY OF THE INVENTION 
     One embodiment of the present invention provides a method for estimating joint load at a joint of a segment. A joint load includes the forces and/or moments acting at a joint. The method comprises the steps of receiving kinematic data, determining a modified acceleration using at least the kinematic data, estimating a joint load using at least the modified acceleration; and determining simulated kinematic data for the segment using at least the joint load. The present invention thus addresses the problems with conventional inverse dynamics analysis by providing a forward dynamics solution for estimation of joint loads that is stable, guaranteed to converge, computationally efficient, and does not require acceleration computations. 
     Another embodiment of the present invention provides a method of predicting simulated kinematic data for a segment. The method comprises the steps of determining a modified acceleration using at least original kinematic data, estimating a joint load for a joint of the segment by using at least the modified acceleration, and predicting kinematic data for the segment based on one or more modified parameters. Therefore, one embodiment of the present invention advantageously allows for the prediction of novel motion. 
     According to one embodiment, modified acceleration may be computed by using one or more error values representing differences between measured kinematic data and previously simulated kinematic data. A feedback gain is applied to force the error values to zero. Therefore, an advantage of the present invention is that the modified acceleration is computed without the need for calculating higher order derivatives of noisy kinematic data, thereby for allowing for estimation or prediction of joint loads without the errors caused by calculation of higher order derivatives. 
     According to further embodiments, the step of estimating a joint load involves recursively estimating the joint loads at successive joints of a three-dimensional system. Recursive embodiments of the present invention are applicable to open chain systems as well as closed chain systems. Therefore, the present invention advantageously eliminates errors caused by parametric uncertainties during estimation of joint forces and moments using closed form, whole body dynamic procedures. According to other embodiments, the step of estimating a joint load involves estimating the joint load using closed form dynamics. Closed form embodiments of the present invention are applicable to open chain systems as well as closed chain systems. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows how a recursive embodiment is used to separate lower body dynamics from upper body dynamics according to one embodiment of the present invention. 
         FIG. 2  is a free body diagram of forces acting on segments in an open chain, three-dimensional serial link system according to one embodiment of the present invention. 
         FIG. 3  is a free body diagram of one segment within a three-dimensional serial link system according to one embodiment of the present invention. 
         FIG. 4  is a block diagram of a tracking system for analysis and synthesis of human motion according to one embodiment of the present invention. 
         FIG. 5  is a flowchart illustrating a tracking process used for a desired time period of human motion analysis according to one embodiment of the present invention. 
         FIG. 6A  is a block diagram of a tracking system where a serial chain system is modeled as an open kinematic chain according to one embodiment of the present invention. 
         FIG. 6B  is a block diagram of a tracking system where a serial chain system is modeled as a closed kinematic chain according to one embodiment of the present invention. 
         FIG. 6C  is a block diagram of a tracking system when kinematic measurements as well as reaction force measurements are available according to one embodiment of the present invention. 
         FIG. 7  is a free body diagram illustrating a four segment, three-dimensional lower extremity model that was used in simulations according to one embodiment of the present invention. 
         FIG. 8  compares simulated ankle joint angle trajectories versus measured ankle joint angle trajectories for a complete cycle of gait according to one embodiment of the present invention. 
         FIG. 9  compares simulated knee joint angle trajectories versus measured knee joint angle trajectories for a complete cycle of gait according to one embodiment of the present invention. 
         FIG. 10  compares simulated hip joint angle trajectories versus measured hip joint angle trajectories for a complete cycle of gait according to one embodiment of the present invention. 
         FIG. 11  compares joint moments at an ankle obtained using traditional inverse dynamics analysis to those obtained using a forward dynamics solution according to one embodiment of the present invention. 
         FIG. 12  compares joint moments at a knee obtained using traditional inverse dynamics analysis to those obtained using a forward dynamics solution according to one embodiment of the present invention. 
         FIG. 13  compares joint moments at a hip obtained using traditional inverse dynamics analysis to those obtained using a forward dynamics solution according to one embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Various embodiments of the present invention provide for estimation of joint loads in multi-dimensional systems, for example estimation of joint loads in human beings. Various embodiments of the present invention also provide for prediction of motions such as prediction of human motion. 
     Various embodiments of the present invention include recursive dynamics, where joint loads are estimated iteratively from one segment to another, as well as closed form dynamics, where all joint loads are computed simultaneously. For example, the selection of a recursive embodiment versus a closed form embodiment may depend upon what sensors are available or on an application for which the invention is being used. 
     Embodiments of the present invention include open-chain systems and closed chain systems. In an open chain system, at most one end of the multi-body system is in contact with the environment. The other end or ends are free or unconstrained. In a closed chain system, more than one end is in contact with the environment. 
     Further, the present invention may be embodied variously using various sensing modalities. In an exemplary sensing modality, only kinematic measurements are available. In another exemplary sensing modality, both kinematic measurements and reaction force measurements are available. 
     I. Recursive Dynamics Embodiments 
     According to one embodiment, joint loads are estimated recursively for a three-dimensional serial chain. A serial chain system includes one or more segments that are connected together by joints, where reaction forces and moments at a joint are shared by the segments connected at the joint. In recursive dynamics, one starts at a first end of a serial chain and calculates successive joint loads moving away from the first end. The joint load estimated for a first joint is used in estimating the joint load for the next joint until the joint or joints in interest are reached. That is, the output of a recursion is force and moment estimation for one or more joints of interest. 
     An advantage of using recursive dynamics is that it allows for estimation of joint loads at a particular joint of interest without the need to model the dynamics of the whole body.  FIG. 1  is an illustration showing how a recursive embodiment is used to separate lower body dynamics from upper body dynamics. The illustration includes upper body portion  105  and lower body portion  110 . A segment of upper body portion  105  is illustrated with load  140 . Lower body portion  110  includes segments having ankle joint  120 , knee joint  125 , and hip joint  130 . In a recursive embodiment for estimating joint forces and moments, the upper body portion  105  can be modeled separately from the lower body portion  110 . Starting with measured reaction forces and moments  115 , for example ground reaction forces and moments, the internal forces and moments acting on ankle joint  120 , knee joint  125 , and hip joint  130  can successively be estimated without considering the effects due to load  140  or the physical parameters of upper body portion  105 , such as mass, center of mass, inertia, and segment dimensions. These parametric uncertainties in upper body portion  105  are significant sources of error in the estimation of the internal forces and moments in the human body when using closed form, whole body dynamic procedures. Although in a particular case it may be desirable to model whole body dynamics, the recursive embodiment provides the flexibility to focus on joints and segments of interest without introducing additional sources of error. 
     Further, measured reaction forces and moments  115  used in recursion provide an extra sensing modality. That is, measured reaction forces and moments  115  provide an additional cue that increases the reliability of the resulting internal force estimations. Human beings may be subjected to unpredictable loads or constrained dynamics resulting from interaction with other objects or other people in the environment. Such circumstances can alter the dynamic representation required to estimate the internal forces and moments at the joints. Some applications for using a recursive embodiment in these circumstances include performing biomechanical studies of lifting tasks and developing controls for assistive devices that aid the physically impaired in everyday tasks. One skilled in the art will appreciate that in-shoe force and pressure sensing devices are complementary technologies that can be used to provide extra sensing modalities for use in various force and moment estimations. 
     A. Recursive Dynamics Method for a Three-Dimensional Serial Chain System 
       FIG. 2  is a free body diagram of forces acting on segments in an open chain, three-dimensional serial link system. The system includes first segment  205 , second segment  210 , and n th  segment  215 . The segments  205 ,  210  . . .  215  are linked by revolute joints. Each of segments  205 ,  210  and  215  is illustrated as a free body diagram in which first joint  220 , second joint  222 , third joint  224 , n th  joint  226  and (n+1) th  joint  228  connect the segments. First segment  205  includes first joint  220  and second joint  222 . Second segment  210  includes second joint  222  and third joint  224 . n th  segment  215  includes n th  joint  226  and (n+1) th  joint  228 . In particular, segments  205  and  210  are connected as follows: second joint  222  links first segment  205  with second segment  210 . Therefore, a serial chain of n segments is formed by connecting adjacent segments at their common or overlapping joint. The joint loads at a joint are shared by the segments connected at the joint. 
     The motion of the three-dimensional serial link system in  FIG. 2  is described by two coordinate systems, a space-fixed or inertial coordinate system, and a moving body-fixed coordinate system that is rigidly fixed to each segment and participates in its motion. The frame descriptions for the inertial coordinate system and the body-fixed coordinate system are denoted by {O} and {B i }, respectively. Each segment&#39;s position relative to frame {O} is described by the vector X i . The notation of vectors written with a leading superscript indicates the coordinate system to which they are referenced. For example,  i N i  represents a vector whose components have numerical values about the axis of frame {B i }. 
     The force due to gravity acting at the i th  segment&#39;s center of mass is denoted by m i g, where m i  is the mass of segment i and g is the 3×1 vector representing the acceleration due to gravity. The vectors  o Γ i  and  i N i  represent the force and moment couple exerted on segment i by segment i−1, with respect to frames {O} and {B i } respectively. For example, with respect to second joint  222 , the force  o Γ 2  and moment  i N 2  exerted by first segment  205  and second segment  210  on each other are illustrated. Similarly, the vector  o Γ i+1  and  i N i+1  represent the force and moment couple exerted on segment i by segment i+1. 
     With reference to  FIG. 2 , an instance of recursive calculation is now described. Recursive analysis begins at n th  segment  215  by determining the forces and moments at n th  joint  226 . The estimated forces and moments at n th  joint  226  are the output of the first step of the recursive analysis. This output is used as input for estimating the forces and moments at the next joint, i.e. the (n-1) th  joint (not shown). In this manner, the recursive analysis of joint forces and moments at successive joints continues until the joint of interest is reached. According to one embodiment, measured ground reaction forces and moments are acting on n th  segment  215 , which is referred to as an end-effector segment, and the forces and moments acting on n th  joint  226  are estimated in terms of the measured ground reaction forces and moments. Next, the forces and moments acting at the (n-1) th  joint are estimated in terms of the previously estimated forces and moments acting at n th  joint  226 . This recursive procedure of using the output of the previous calculation as an input for the current calculation is repeated until forces and moments have been estimated for the joint or joints of interest. One skilled in the art will appreciate that first segment  205  is not necessarily an end segment in a multi-segment system. Rather, first segment  205  is the segment at which it is desired to stop the recursive computation if one is interested in obtaining estimates of the forces and moments at first joint  220 . It should be further noted that measured reaction forces and moments  115  ( FIG. 1 ) act at a point of contact with the environment, which is not necessarily a joint. 
       FIG. 3  is a free body diagram of one segment within a three-dimensional serial link system. i th  segment  300  is one segment within a three-dimensional serial link system, such as the system illustrated in  FIG. 2 . As illustrated in  FIG. 3 , i th  segment  300  includes joint i  305  and joint i+1  310 . The origin of the frame {B i } represents the position of the i th  segment&#39;s center of mass  315 . The position of i th  segment&#39;s center of mass  315  relative to frame {O} is described by vector X i . The joint centers at joint i  305  and joint i+1  310  are described by the vectors C i  and C i+1  respectively, with respect to frame {O}. The position vectors in frame {O} from the i th  segment&#39;s center of mass  315  to the connection points C i  and C i+1 , are described by L i,i  and L i,i+1 , respectively. The force due to gravity acting at the i th  segment&#39;s center of mass  315  is denoted by m i g, where mi is the mass of segment i and g is the 3×1 vector representing the acceleration of gravity. The vectors  o Γ i  and N i  represent the force and moment couple exerted on segment i by segment i−1, with respect to frames {O} and {B i }, respectively. Similarly, the vector  o Γ i+1  and  i N i+1  represent the force and moment couple exerted on segment i by segment i+1, with respect to frames {O} and {B i }, respectively. 
     B. Description of Rotation Transforms 
     The orientation of frame {B i } relative to frame {O} is achieved through orthogonal matrix  o R i , where the notation with the relative transformation is with respect to the frame with the leading superscript. The rotation matrix represents three consecutive rotations using Euler angles Θ=[ø, θ, ψ] T . Each rotation is performed about an axis of the moving reference frame. The explicit expression for the rotation matrix is given by equation 1 below, where the notations c and s represent sin and cos, respectively. 
     
       
         
           
             
               
                 
                   
                     R 
                     i 
                       
                       
                       
                     o 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                             - 
                             
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 c 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                             - 
                             
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         
                           
                             
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                             + 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 s 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                             + 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 c 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The inverse transformation from frame {O} to frame {B i } is give by equation 2 below.
 
 i R o = o R i   −1 = o R i   T   (2)
 
     The Euler angles and their derivatives in terms of the angular velocity W=[w x , w y , w z ] T  along the body fixed axes is given by equation 3 below.
 
{dot over (Θ)}=H −1 W  (3)
 
     H −1  is defined in additional detail by equation 4 below. 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       - 
                       1 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           
                             
                               - 
                               s 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               - 
                               s 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                             
                               - 
                               c 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The inverse transformation is given by equation 5 below.
 
W=H{dot over (Θ)}  (5)
 
     H is defined in additional detail by equation 6 below: 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           
                             
                               - 
                               s 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The angular acceleration {dot over (W)} can be obtained by differentiating angular velocity W in equation 5 with respect to time. The angular acceleration {dot over (W)} is given by equation 7 below.
 
 {dot over (W)}=H{umlaut over (Θ)}+{dot over (H)}{dot over (Θ)}   (7)
 
     {dot over (H)} is defined in additional detail by equation 8 below. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         H 
                         . 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       ψ 
                                       ⁢ 
                                       
                                           
                                       
                                     
                                     . 
                                   
                                   ⁢ 
                                   s 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       θ 
                                       ⁢ 
                                       
                                           
                                       
                                     
                                     . 
                                   
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     s 
                                     ⁡ 
                                     
                                       ( 
                                       ψ 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     
                                       ψ 
                                       ⁢ 
                                       
                                           
                                       
                                     
                                     . 
                                   
                                 
                                 ⁢ 
                                 
                                   s 
                                   ⁡ 
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   
                                     ψ 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     s 
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                   ⁢ 
                                   s 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     θ 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     c 
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     ψ 
                                     . 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   c 
                                   ⁡ 
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     . 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                             
                               
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     C. Derivation of Newton-Euler Equations 
     The Newton-Euler equations of motion are derived using special notation where the translational and rotational equations of motion are combined in vector and matrix form. Although, the equations are derived for isolated i th  segment  300 , which forms a serial chain with its two neighboring segments, those skilled in the art will recognize that the following derivation can be extended and recursively applied in any serial chain or branching chain mechanism with one or more segments. Further, consistent with most models of human anatomy, each joint is assumed to have three translational and three rotational degrees of freedom. Those skilled in the art will recognize that the following derivation is also applicable to other joint models. 
     For isolated i th  segment  300 , the Euler angles relative to frame {O} are given by Θ i =[ø i , θ i , ψ i ] T . The angular velocity for i th  segment  300  is represented by W i =[w ix , w iy , w iz ] T , and the angular acceleration for i th  segment  300  is represented by {dot over (W)} i . The position of the center of mass  315  of the i th  segment  300  relative to frame {O} is described by vector X i , the linear acceleration of i th  segment  300  is given by {umlaut over (X)} i , and the linear velocity of i th  segment  300  is given by {dot over (X)} i . 
     The Newton-Euler state space equations describing the motion of i th  segment  300  about its center of mass  315  are set forth below in equation 9, equation 10 and equation 11. See Hemami, A State Space Model for Interconnected Rigid Bodies, IEEE Trans. on Automatic Control, 27(2): 376-382, 1982, which is incorporated by reference herein in its entirety. I is the inertia tensor of i th  segment  300  in frame {B i }, and f(W)=W×I W is the “gyroscopic torque”.
 
 m   i   {umlaut over (X)}   i = o Γ i − o Γ i+1   −m   i   g   (9)
 
{dot over (Θ)} i   =H   i   −1   W   i   (10)
 
 I   i   {dot over (W)}   i =−ƒ( W   i )+ i   N   i − i   N   i+1 + i   R   o ( L   i,i × o Γ i )− i   R   o ( L   i,i+1 × o Γ i+1 )  (11)
 
     One skilled in the art will appreciate that equation 9 represents an expression for summing the translational forces acting on segment  300 , equation 10 represents the angular velocity of i th  segment  300 , and equation 11 represents an expression for summing the torques acting at joint i  305  and joint i+1  310 . 
     To simplify notation and avoid using cross products, consider the following identity for cross product of arbitrary vectors a and b, where the tilde symbol ({tilde over ( )}) denotes the skew symmetric matrix representation of a vector.
 
 a×b=ãb   (12)
 
     For example, with the vector L=[l x l y l z ] T  one can associate the 3×3 skew symmetric matrix {tilde over (L)}, defined by equation 13 below. 
     
       
         
           
             
               
                 
                   
                     L 
                     ~ 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           
                             - 
                             
                               l 
                               z 
                             
                           
                         
                         
                           
                             l 
                             y 
                           
                         
                       
                       
                         
                           
                             l 
                             z 
                           
                         
                         
                           0 
                         
                         
                           
                             - 
                             
                               l 
                               x 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               l 
                               y 
                             
                           
                         
                         
                           
                             l 
                             x 
                           
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The representation of equation 11 with the new notation is expressed in equation 14 below.
 
 I   i   {dot over (W)}   i =−ƒ( W   i )+ i   N   i − i   N   i+1 + i   R   o   {tilde over (L)}   i,i   o Γ i − i   R   o   {tilde over (L)}   i,i+1   o Γ i+1   (14)
 
     Making use of equation 7 above and transforming  i N i  and  i N i+1  to frame {O}, equation 14 is expressed as equation 15 below.
 
 I   i   H   i {umlaut over (Θ)} i   =−f ( W   i )− I   i   {dot over (H)}   i {dot over (Θ)} i + i   R   o   o   N   i − i   R   o   o   N   i+1 + i   R   o   {tilde over (L)}   i,i   o Γ i − i   R   o   {tilde over (L)}   i,i+1   o Γ i+1   (15)
 
     Combing the translational equations of motion, equation 9, and the rotational equations of motion, equation 15, we obtain a compact representation in matrix form of the Newton-Euler equation for an isolated body segment, which is set forth in equation 16 below. Vector q i =[X i Θ i ] T  represents the coordinates of the center of mass  315  of i th  segment  300  and the Euler angles of the i th  segment  300 . One skilled in the art will appreciate that the term {umlaut over (q)} i  in equation 16 represents the second derivative of vector q i , and {dot over (q)} i  in equation 16 represents the first derivative of vector q i .
 
 M   i ( q   i ){umlaut over (q)} i   =P   i ( q   i   , {dot over (q)}   i )+ A   p     i   ( q   i ) U   i   +A   d     i   ( q   i ) U   i+1   (16)
 
     The individual elements of equation 16 are defined in additional detail in equations 17 through 22. In equation 16, the notation Ī represents the 3×3 identity matrix, and Ø represents the 3×3 matrix with all zero elements. U i  is a vector whose elements represent the forces and moments acting at joint i  305  of i th  segment  300 , and U i+1  is a vector whose elements represent the forces and moments acting at joint i+1  310  of i th  segment  300 . 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         q 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 m 
                                 i 
                               
                               ⁢ 
                               
                                 I 
                                 - 
                               
                             
                           
                         
                         
                           
                             ∅ 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∅ 
                           
                         
                         
                           
                             
                               
                                 I 
                                 i 
                               
                               ⁢ 
                               
                                 H 
                                 i 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         A 
                         d 
                       
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         q 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               - 
                               
                                 I 
                                 - 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   R 
                                   o 
                                     
                                     
                                     
                                   i 
                                 
                               
                               ⁢ 
                               
                                 
                                   L 
                                   ~ 
                                 
                                 
                                   i 
                                   , 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∅ 
                           
                         
                         
                           
                             
                               - 
                               
                                 R 
                                 o 
                                   
                                   
                                   
                                 i 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         A 
                         p 
                       
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         q 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               I 
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 R 
                                 o 
                                   
                                   
                                   
                                 i 
                               
                               ⁢ 
                               
                                 
                                   L 
                                   ~ 
                                 
                                 
                                   i 
                                   , 
                                   i 
                                 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∅ 
                           
                         
                         
                           
                             
                               R 
                               o 
                                 
                                 
                                 
                               i 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       P 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           q 
                           i 
                         
                         , 
                         
                           
                             q 
                             . 
                           
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   - 
                                   
                                     m 
                                     i 
                                   
                                 
                                 ⁢ 
                                 g 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   I 
                                   i 
                                 
                               
                               ⁢ 
                               
                                 
                                   H 
                                   . 
                                 
                                 i 
                               
                               ⁢ 
                               
                                 
                                   q 
                                   . 
                                 
                                 i 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 f 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     W 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     U 
                     i 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Γ 
                             i 
                               
                               
                               
                             o 
                           
                         
                       
                       
                         
                           
                             N 
                             i 
                               
                               
                               
                             o 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     U 
                     
                       i 
                       + 
                       1 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Γ 
                             
                               i 
                               + 
                               1 
                             
                               
                               
                               
                             o 
                           
                         
                       
                       
                         
                           
                             N 
                             
                               i 
                               + 
                               1 
                             
                               
                               
                               
                             o 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     D. Inverse Dynamics Problem 
     In inverse dynamics analysis, forces and moments acting at joints are computed from measured or desired kinematic data. One skilled in the art will recognize that statistical or geometric approaches operating on measured marker positions may be used to obtain the measured generalized coordinates for each segment, which are represented as q m     i   , where i is the segment number. As explained above, q m     i    represents the three coordinates and the three Euler angles for i th  segment  300 . The vector of all generalized coordinates from segment i to segment n is expressed as q m  in equation 23 below.
 
q m =[q m     i     T q m     i+1     T  . . . q m     n     T ] T   (23)
 
     Referring to  FIG. 2 , an inverse dynamics analysis may be performed based on inward recursions starting from n th  segment  215  and working inward toward the joint of interest, for example i th  joint  305 . Pursuant to an inverse dynamics analysis, the forces and moments at joints n through i can be computed using recursion equation 24 below, which has been derived from equation 16 above. According to one embodiment, the measured generalized coordinates for each segment, q m , as well as their derivatives are known, as are the measured reaction forces and moments acting on n th  segment  215 , which are represented as U n+1 . These known parameters are used as inputs in recursion equation 24.
 
 U   i   =A   p   −1 ( q   m     i   ){ M   i ( q   m     i   ) {umlaut over (q)}   i   −P   i ( q   m     i     , {dot over (q)}   m     i   )− A   d     i   ( q   m     i   ) U   i+1 }  (24)
 
     The inverse dynamics analysis allows us to calculate the vector U, shown in equation 25 below, of forces and moments at joints n through i.
 
U=[U i   T  U i+1   T  . . . U n   T ] T   (25)
 
     E. Forward Dynamics Solution 
     Various embodiments of the present invention provide a control theoretic framework for analysis and synthesis of human motion, whereby estimation and synthesis of human motion has been formulated as a trajectory tracking control problem. Various embodiments of the present invention provide a tracking controller whose output represents the forces and moments that when applied to a forward dynamics module will reproduce or “track” the measured kinematic data. The tracking controller estimates the joint loads necessary to force a biomechanical system to track measured kinematics. The desired tracking is achieved by employing a nonlinear control law that linearizes and decouples every state such that the tracking error can be forced to zero. Tracking methods have been previously used in the robotics community for the purpose of manipulator control, as explained in Craig, Introduction to Robotics, Mechanics and Control, Addison-Wesley, 2 nd  ed., 1989, which is hereby incorporated by reference in its entirety. Various embodiments of the present invention apply the concept of feedback linearization to estimate joint torques for human motion analysis and synthesis. Various embodiments of the present invention apply the concept of feedback linearization to predict human motions. Embodiments of the tracking controller for closed form computations are presented below. Embodiments of the tracking controller for recursive estimation are now explained with reference to  FIG. 4 . 
       FIG. 4  is a block diagram of one embodiment of a tracking system for analysis and synthesis of human motion. With reference to an exemplary three-dimensional serial chain system in  FIG. 2 , the tracking system of  FIG. 4  provides for estimation of joint loads at a particular time by using the following information: simulated kinematic data in the form of simulated state variables that were calculated using joint loads estimated for some previous instant in time; and measured kinematic data as well as their derivatives at the particular time. The tracking system in  FIG. 4  comprises tracking controller  410  and forward dynamics module  425 . Tracking controller  410  comprises error correction controller  415  and inverse dynamics module  420 . 
     Error Correction Controller  415   
     Inputs to error correction controller  415  include measured kinematics  450 , including measured positional data q m , and measured velocity data {dot over (q)} m  that are estimated by numerical differentiation of the kinematic data. Acceleration kinematic data {umlaut over (q)} m , which are estimated by numerical differentiation of measured velocity data, can be used in noise free applications, but are not required. For an exemplary three-dimensional serial chain system having n segments, measured kinematic data q m  represents an n*6 matrix that provides the three measured Euler angles and three measured center of mass coordinates for each segment. Velocity kinematic data {dot over (q)} m  represents an n*6 matrix that provides measured velocities of the three Euler angles and three center of mass coordinates for each segment. Similarly, acceleration kinematic data {umlaut over (q)} m  represents an n*6 matrix that provides accelerations of the three Euler angles and three center of mass coordinates for each segment. In  FIG. 4 , measured kinematics  450  represent the kinematics for each segment at a present instant in time for which tracking is being performed. 
     According to one embodiment, simulated kinematic data comprises simulated state variables q and {dot over (q)}, and inputs to error correction controller  415  also include simulated state variables q and {dot over (q)} that are obtained as a result of forward dynamics simulations. For an exemplary three-dimensional serial chain system having n segments, simulated positional data q  470  represents an n*6 matrix that provides the three simulated Euler angles and three simulated center of mass coordinates for each segment. Similarly, simulated velocity data {dot over (q)}  475  represents an n*6 matrix that provides simulated velocities of the three Euler angles and three center of mass coordinates for each segment. 
     Simulated positional data q  470  represents simulated positional data for the present instant in time for which tracking is being performed, and is calculated using joint loads  480 , simulated state variables and (optionally) measured reaction forces and moments  460  at a previous instant in time, as explained below. Simulated velocity data {dot over (q)}  475  represents simulated velocity data for the present instant in time for which tracking is being performed, and is calculated using joint loads  480 , simulated state variables and (optionally) measured reaction forces and moments  460  at a previous instant in time, as explained below. 
     In the illustrated embodiment, error correction controller  415  comprises modules that implement equation 26 below. For each segment, error correction controller  415  compares measured positional data at the present time instant to simulated positional data  470  for the present time instant to obtain a positional error signal e. For each segment, positional error signal e represents positional errors for the three Euler angles and the three coordinates of the center of mass. Similarly, error correction controller  415  compares measured velocity data at the present time to simulated positional data  470  for the present time to obtain a velocity error signal ė. For each segment, velocity error signal ė represents velocity errors for the three Euler angles and the three coordinates of the segment&#39;s center of mass. 
     The error signals are modulated by position and velocity feedback gain matrices K p     i   , and K v     i    and summed to arrive at an acceleration term {umlaut over (q)} i * that is an estimate (or modification) of the accelerations of the generalized coordinates. The modified acceleration term for each segment at the present instant in time is given by equation 26 below.
 
 {umlaut over (q)}   i   *=a {umlaut over (q)}   m     i     +K   p     i   ( q   m     i     −q   i )+ K   v     i   ( {dot over (q)}   m     i     −{dot over (q)}   i )  (26)
 
     The feedback matrices K v     i    and K p     i    are diagonal matrices, the details of which are chosen based on experimental observation to achieve a critically damped response. The relation between K v  and K p  to achieve a critically damped response is given by equation 27 below.
 
K v =2√{square root over (K p )}  (27)
 
     The parameter a is a scalar parameter that may be set to zero or one. According to one embodiment, a=1 and measured accelerations are used to compute the modified acceleration term. According to another embodiment, a=0 and measured accelerations are ignored. One skilled in the art will appreciate that when parameter a is set equal to zero the second derivative of the kinematic data, the estimated accelerations term {umlaut over (q)} m     i   , is ignored in equation 26. Therefore, in such a case, only modified acceleration {umlaut over (q)} i * is used in the tracking system. An advantage of not using the second derivative of noisy kinematic data is improved accuracy of force and moment estimation. 
     The modified acceleration for each segment in a three-dimensional system is concisely represented by vector array {umlaut over (q)}*, as shown in equation 28 below. Error correction controller  415  outputs modified acceleration {umlaut over (q)}* as an input to inverse dynamics module  420 .
 
{umlaut over (q)}*=[{umlaut over (q)} i * T {umlaut over (q)}   i+1 * T  . . . {umlaut over (q)} n * T ] T   (28)
 
     Inverse Dynamics Module  420   
     In addition to modified acceleration {umlaut over (q)}*, inputs to inverse dynamics module  420  include simulated state variables q and {dot over (q)}. Measured reaction forces and moments  460  may optionally be included as additional inputs to inverse dynamics module  420 . According to one embodiment, measured reaction forces and moments comprise ground reaction forces and moments at the constrained end of a three-dimensional serial chain system. According to an exemplary embodiment, joint loads (U)  480  at the present instant in time are estimated using inverse dynamics module  420 . The vector U represents a matrix of joint forces and moments at one or more joints of an exemplary serial chain system. The joint loads at an exemplary i th  joint may be represented as U i . According to one embodiment, the joint loads U i  that will drive body segment i to follow the desired trajectory q m     i    (obtained from measurements) is obtained by the control law in equation 29 below. The structure of equation 29 differs from the inverse dynamics solution in equation 24 in that the input parameters are functions of measured feedback rather than measured kinematics data.
 
 U   i   =A   p   −1 ( q   i ){ M   i ( q   i ) {umlaut over (q)}   i   *−P   i ( q   i   , {dot over (q)}   i )− A   d     i   ( q   i ) U   i+1 }  (29)
 
     According to one embodiment, for each time instant for which inverse dynamics analysis is being performed, inverse dynamics module  420  recursively applies the control law in equation 29 to estimate the joint loads at successive joints of a serial chain system. As seen in equation 29, inverse dynamics module  420  uses the joint loads U i+1  at an (i+1) th  joint to estimate the joint loads U i  at a neighboring i th  joint. In this manner, inverse dynamics module  420  recursively estimates the joint loads from an n th  joint of an end effector segment until the joint loads at an i th  joint of interest have been determined. According to another embodiment, joint loads (U)  480  at one or more joints are estimated using closed form dynamics, as explained in more detail below. The output of inverse dynamics module  420 , representing the estimated joint loads at joints i . . . n is given by equation 30 below.
 
U=[U i   T  U i+1   T  . . . U n   T ] T   (30)
 
     Forward Dynamics Module  425   
     Inputs to forward dynamics module  425  include joint loads U for the present instant in time for which analysis is being performed. Additionally, the simulated state variables q and {dot over (q)} for the present instant in time are also used as input by forward dynamics module  425 . According to one embodiment, when the tracking system in  FIG. 4  is applied to obtain estimated joint loads (U)  480 , the parameters for the forward dynamics module  425  (such as P i , M i , A p , and A d     i   ) are identical to the parameters used by inverse dynamics module  420 . 
     Forward dynamics module  425  computes induced accelerations {umlaut over (q)} i  at an exemplary i th  joint of interest, for the present time instant for which analysis is being performed, according to equation 31 below. The vector containing the induced accelerations at joints i . . . n is given by equation 32 below.
 
 {umlaut over (q)}   i   =M   i   −1 ( q   i )[ P   i ( q   i   , {dot over (q)}   i )+ A   p     i   ( q   i )+ A   d     i   ( q   i ) U   i+1 ]  (31)
 
{umlaut over (q)}=[{umlaut over (q)} i   T  {umlaut over (q)} i+1   T  . . . {umlaut over (q)} n   T ] T   (32)
 
     Matrices M i , P i , A p     i    and A d     i    have been defined above in equations 17 through 20. Vectors L i,i  and L i,i+1  in these matrices are computed according to equations 33 and 34 below, where the measured joint centers, C m     i    and C m     i     +1  , are used as constraints to avoid numerical drift. One skilled in the art will recognize that the measured joint centers are derived from measured kinematics and anthropometric measurements. According to one embodiment, the measured joint centers may be obtained statistically based on regression equations. According to another embodiment, the measured joint centers may be obtained geometrically based on functional methods.
 
 L   i,i   =C   m     i     −X   i   (33)
 
 L   i,i+1   =C   m     i+1     −X   i   (34)
 
     Having obtained the induced acceleration for the present time instant, t, for which analysis is being performed, forward dynamics module  425  performs a simulation to determine state variables q i  and {dot over (q)} i  for a next instant in time, t+Δt, based on available estimated joint loads U  480  and available simulated state variables for the present instant in time. According to one embodiment, forward dynamics module  425  performs numerical integration of the induced accelerations for each segment to obtain the simulated state variables (simulated positional and velocity data) for each segment at the next instant in time for which analysis is being performed, i.e. at t+Δt. In one embodiment, numerical integration is performed by an integration function in MATLAB software that is commercially available from The MathWorks, Inc., Natick, Massachusetts. One skilled in the art will appreciate that integration can be performed by many methods, such as the Runge Kutta method. 
     As described above, forward dynamics module  425  obtains simulated state variables q and {dot over (q)} for the next instant in time by numerical integration of the induced accelerations for the present instant in time. The state variables q and {dot over (q)} are input to error correction controller  415 , to be used in determining modified acceleration {umlaut over (q)}* for the next time step. Error correction controller  415  generates modified accelerations {umlaut over (q)}* such that the inverse dynamics module  420  computes a set of inputs or controls denoted by U that when applied to forward dynamics module  425  substantially reproduces or tracks the measured kinematics  450 . 
     For a desired time period of human motion analysis, the tracking system in  FIG. 4  forces the tracking error between simulated and measured kinematics to approach zero as time progresses, thereby ensuring the accuracy of estimated joint loads. According to one embodiment, the system is initialized by setting the simulated state variables at time t=0 equal to the measured kinematics  450  at t=0. 
     One skilled in the art will appreciate that the described equations, expressions, modules, or functions can be implemented in a general-purpose computer, special purpose computer, or hardware. In an embodiment, a software programmable general-purpose computer implements features of the invention. The software is preferably distributed on a computer readable medium, which includes program instructions. A computer readable medium includes, for example, a computer readable storage volume. The computer readable storage volume can be available via a public computer network, a private computer network, or the Internet. One skilled in the art will appreciate that the program instructions can be in any appropriate form, such as source code, object code, or scripting code. 
       FIG. 5  is a flowchart illustrating a tracking process used for a desired time period of human motion analysis according to one embodiment of the present invention. The flowchart in  FIG. 5  is explained with reference to an exemplary three-dimensional serial chain system described in  FIG. 2  and with reference to an exemplary i th  segment  300  described in  FIG. 3 . The start  500  of the tracking process occurs at an initial time, t=0, chosen as the time to start estimating joint loads at joints of one or more segments. The process starts by obtaining initial conditions  505  that represent initial values of simulated state variables q and {dot over (q)}. According to an exemplary embodiment, the initial conditions are obtained from measured kinematic data by setting the initial values of simulated kinematics to be equal to the initial values of measured kinematics, as shown in equations 35 and 36 below.
   q ( t =0)= q   m ( t =0)  (35)   {dot over (q)} ( t =0)= {dot over (q)}   m ( t =0)  (36) 
     For a present instant in time at which a serial chain system&#39;s motion is being analyzed, the process obtains measured kinematics  510  for each segment in the chain. One skilled in the art will recognize that statistical or geometric approaches operating on measured marker positions may be used to obtain the measured generalized coordinates for each segment, which are represented as q m . Velocity kinematic data {dot over (q)} m , and acceleration kinematic data, {umlaut over (q)} m , are estimated by numerical differentiation of the kinematic data. Next, the process computes velocity tracking error, ė,  515  for each segment at the present instant in time. For each segment, velocity tracking error ė is computed by comparing velocity kinematic data at the present time instant to simulated velocity data  475  for the present time instant. As described above, velocity error signal ė represents velocity errors for the three Euler angles and the three coordinates of the center of mass. Similarly, the process computes positional tracking error, e,  520  for each segment at the present instant in time. For each segment, positional tracking error e is computed by comparing measured positional kinematic data at the present time instant to simulated positional data  470  for the present time instant. As described above, positional error signal e represents positional errors for the three Euler angles and the three coordinates of the center of mass. 
     The process computes modified acceleration  525  for each segment at the present time instant using equation 26 above. According to one embodiment, the scalar a is set to zero in equation 26, thereby ignoring acceleration kinematic data during the tracking process and improving the accuracy of force and moment estimations. Modified accelerations computed at step  525  are used to perform inverse dynamics analysis  530 , thereby obtaining estimated forces and moments at each joint for the present instant in time. According to one embodiment, the step of performing inverse dynamics analysis  530  recursively estimates the forces and moments at each joint of a serial chain system according to equation 29. According to another embodiment, the step of performing inverse dynamics analysis uses closed form analysis to estimate the forces and moments at each joint of a serial chain system. Further details of a closed form embodiment are explained below. 
     At step  535 , the process performs forward dynamics analysis to obtain the simulated state variables (simulated kinematic data) for each segment at the next instant in time for which human motion analysis is desired. Forward dynamics analysis  535  has been explained in further detail above with reference to forward dynamics module  425 . 
     Once simulated kinematics have been obtained, at step  540  the process determines whether the desired time period for human motion analysis has ended, i.e. whether t=t final ? If not, then the process selects the next instant in time for which joint loads are to be estimated by incrementing the present time instant, t, by Δt at step  545 , followed by repeating steps  510  through  540  of the tracking process described above to estimate the forces and moments at the joints of the serial chain system at the newly selected instant in time. If the desired time period for human motion analysis has ended, then the process ends at step  550 . 
     F. Closed Loop Error Dynamics 
     Various embodiments of the present invention provide a tracking system for analysis and synthesis of human motion that employs a feedback control law described by equation 29 above. To demonstrate the tracking performance of this system, it is instructive to consider the closed loop error dynamics. The closed loop response and the associated error dynamics of tracking controller  410  may be described by a linear system. As shown in equation 37 below, positional error signal e i  denotes the error between the measured kinematics q m     i    and the simulated state variable q i , which is obtained by integration in forward dynamics module  425 . The error dynamics for several scenarios are described below.
 
 e   i   =q   m     i     −q   i   (37)
 
     Accelerations are Included: a=1 
     In the ideal situation of perfect measurements and zero error in numerical differentiation, the closed loop error dynamics is given by differential equation 38 below.
 
 ë   i   +K   v     i     ė   i   +K   p     i     e   i =0  (38)
 
     The error dynamics of state variable q i  can be independently controlled by eigenvalue assignment. Let λ 1  and λ 2  denote the eigenvalues of the above differential equation 38. Equation 39 provides a critically damped solution, i.e. no sinusoidal oscillations, with real and equal eigenvalues. This solution yields the fastest non-oscillatory response. As described above, the relationship between K p  and K v  to achieve a critically damped response is set forth in equation 27 above.
 
 e ( t )= c   1   e   λ     1     t   +c   2    te   λ     2     t   (39)
 
     Accelerations Ignored: a=0 
     Suppose the measured accelerations that are estimated from the measured kinematics are ignored by setting a=0. The closed loop error dynamics is expressed by non-homogeneous differential equation 40 below.
 
 ë   i   +K   v     i     ė   i   +K   p     i     e   i   ={umlaut over (q)}   m     i     (40)
 
     Although the solution to the above differential equation 40 contains a forcing term, assuming the acceleration term {umlaut over (q)} m     i    is bounded, the error will converge to zero by assigning the eigenvalues of equation 40 to have negative and real parts. As before, the gains K p     i    and K v     i    may be appropriately designed for a critically damped response using the relation given in equation 27 above. 
     Incorporating Derivative Estimation Error 
     In the above formulation of the error equations, it was assumed that the velocity kinematic data {dot over (q)} m     i    and accelerations kinematic data {umlaut over (q)} m     i    can be precisely calculated by differentiating the measured positional data q m     i   . Indeed, errors in numerical differentiation of noisy kinematic measurements cannot be ignored and are considered in the following formulation. 
     Let              v  and            a  represent the bounded error in the velocity and acceleration calculations. The estimates
                 q   .     ^       m   i           
and
 
                 q   ..     ^       m   i           
may be expressed by equations 41 below.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           q 
                           . 
                         
                         ^ 
                       
                       
                         m 
                         i 
                       
                     
                     = 
                     
                       
                         
                           q 
                           . 
                         
                         
                           m 
                           i 
                         
                       
                       + 
                       
                         ɛ 
                         v 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           q 
                           .. 
                         
                         ^ 
                       
                       
                         m 
                         i 
                       
                     
                     = 
                     
                       
                         
                           q 
                           .. 
                         
                         
                           m 
                           i 
                         
                       
                       + 
                       
                         ɛ 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     The closed loop dynamics incorporating the derivative estimation error is given by equation 42 below. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       ⁢ 
                       
                         
                           
                             q 
                             .. 
                           
                           ^ 
                         
                         
                           m 
                           i 
                         
                       
                     
                     - 
                     
                       
                         q 
                         .. 
                       
                       i 
                     
                     + 
                     
                       
                         K 
                         
                           v 
                           i 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 q 
                                 . 
                               
                               ^ 
                             
                             
                               m 
                               i 
                             
                           
                           - 
                           
                             
                               q 
                               . 
                             
                             i 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         K 
                         
                           p 
                           i 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             q 
                             
                               m 
                               i 
                             
                           
                           - 
                           
                             q 
                             i 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     Substituting equations 41 into equation 42, equation 43 below is derived. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       ⁢ 
                       
                         
                           
                             q 
                             ¨ 
                           
                           ^ 
                         
                         
                           m 
                           i 
                         
                       
                     
                     - 
                     
                       
                         q 
                         ¨ 
                       
                       i 
                     
                     + 
                     
                       
                         K 
                         
                           v 
                           i 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               q 
                               . 
                             
                             
                               m 
                               i 
                             
                           
                           - 
                           
                             
                               q 
                               . 
                             
                             i 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         K 
                         
                           p 
                           i 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             q 
                             
                               m 
                               i 
                             
                           
                           - 
                           
                             q 
                             i 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     - 
                     
                       ( 
                       
                         
                           a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ɛ 
                             a 
                           
                         
                         + 
                         
                           
                             K 
                             
                               v 
                               i 
                             
                           
                           ⁢ 
                           
                             ɛ 
                             v 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     The error dynamics for a=0 and a=1 are given by equations 44 and 45 below, respectively.
 
 ë   i   +K   v     i     ė   i   +K   p     i     e=−K   v     i   ε v   +{umlaut over (q)}   m     i      a =0  (44)
 
 ë   i   +K   v     i     ė   i   +K   p     i     e =−(ε a   +K   v     i   ε v )  a =1  (45)
 
     G. Open Chain and Closed Chain Embodiments 
     Recursive embodiments of the present invention include both open chain and closed chain estimations. An open chain system is constrained with the environment at one end, while the remaining terminal segments are free. A closed chain system has more than one end in contact with the environment. A recursive embodiment has been described with reference to  FIG. 2  above, wherein the joint loads at successive joints are computed recursively starting with an n th  joint  226  and ending with a joint of interest, such as the first joint  220 . An advantage of a recursive formulation is that the entire body need not be modeled. The force and moment estimation is complete at the segment of interest regardless of whether the segment of interest is the last segment of the serial system. As described above with reference to  FIG. 1 , the parametric uncertainties in upper body portion  105  are significant sources of error in the estimation of internal forces and moments. The uncertainties can be avoided, however, when only joint moments proximal to the force plate are desired. 
     In one recursive embodiment of the present invention, open chain estimation is performed. In an open chain system, the one end in contact with the environment is termed a constrained end. In an embodiment of the present invention, the constrained end is a human being&#39;s foot that is in contact with the ground or other supporting surface. 
     In another recursive embodiment of the present invention, closed chain estimation is performed. A closed chain system has more than one end in contact with the environment. Referring to  FIG. 2 , with force plate measurements for n th  segment  215  available, the joint loads at the joint of interest are computed recursively with equation 29 above, starting with n th  joint  226  and working towards the joint of interest. 
     II. Multi-modal Embodiments 
     Since the optimal representation of the dynamic equations of motion will differ depending on the available sensors, inverse dynamics is in general considered a multi-modal sensing problem. See Dariush, Hemami and Parnianpour, Multi-modal Analysis of Human Movement from External Measurements, Journal of Dynamic Systems, Measurement and Control, 123(2): 272-278, 2002, which is incorporated by reference herein in its entirety. The tracking system of the present invention may be embodied using various modes of sensing.  FIG. 6  depicts two exemplary modes of sensing having three different system level structures. Tracking error  605  in  FIG. 6  represents positional tracking error e and velocity tracking error ė. 
       FIGS. 6A and 6B  illustrate block diagrams of embodiments of a tracking system when only measured kinematics  450  are available.  FIG. 6A  illustrates an embodiment where a serial chain system is modeled as an open-kinematic chain, such as the single support phase of gait. For an open chain system, reaction forces and moments need not be estimated or measured for joint load estimation or for prediction of simulated kinematics  610 .  FIG. 6B  illustrates an embodiment when a serial chain system is modeled as a closed kinematic chain, such as the double support phase of gait. Open and closed chain embodiments of the tracking system are described in more detail below. For closed chain systems, reaction forces and moments need to be estimated or measured for joint load estimation or for prediction of simulated kinematics  610 . As illustrated in  FIG. 6B , constraints module  615  is used to estimate reaction forces  620 , which are inputted into forward dynamics module  425 . According to one embodiment of the present invention, reaction forces  620  are analytically estimated using the joint load, and previously simulated kinematic data simulated at a previous time. For example, the previously simulated kinematic data is simulated at the previous time-step. 
       FIG. 6C  illustrates a block diagram of an embodiment of the tracking system when measured kinematics  450  as well as measured reaction forces and moments  460  are available, which may be obtained, for example, from motion capture and force plate measurements. 
     The choice between a recursive dynamics method versus a closed form dynamics method depends on the available sensing modality and the specific application. For example, closed form dynamics is well suited for joint load estimation or prediction of simulated kinematics  610  for the tracking system illustrated in  FIG. 6B , which involves a closed chain system with only measured kinematics  450  available. On the other hand, the recursive dynamics method is less sensitive to parametric uncertainties in a serial chain system. The recursive dynamics method is well suited for joint load estimation or prediction of simulated kinematics  610  for the tracking systems illustrated in  FIG. 6C , which involves an open or closed chain system with both measured or estimated kinematics as well as reaction forces and moments  460  available. 
     III. Closed Form Dynamics Embodiments 
     Various embodiments of the present invention express the equations of motion in closed form. For example, closed form embodiments are useful for analysis of an open or closed kinematic chain with only motion capture data available as an input. Derivation of the closed form dynamic equations for the purpose of tracking control are well studied in the robotics literature and one skilled in the art will recognize that those equations can readily be applied for human motion analysis and synthesis according to the tracking system of the present invention. Examples of derivation of closed form dynamics include Newtonian Mechanics, Lagrangian Formulation, Kane&#39;s method, Hamiltonian method. Closed form embodiments include open kinematic chains as well as closed kinematic chains. 
     A. Closed Form Dynamics Embodiments With Open Kinematic Chains 
     In closed form embodiments modeled as open kinematic chains, a full kinematic description obtained from motion capture is sufficient to estimate or predict joint loads. An open kinematic chain can be modeled in closed form using differential equation 46 below.
 
 M ( q ) {umlaut over (q)}+H ( q,{dot over (q)} ) {dot over (q)}+G ( q )= U   (46)
 
     In equation 46, vector q represents the measured coordinates, M is the inertia matrix, H describes the effect of coriolis and centrifugal forces, and G is the vector of gravitational forces. The vector U represents joint loads. Equation 46 and all its component vectors and matrices are described in detail in J. Craig, Introduction to Robotics: Mechanics and Control, 2 nd  Edition, Addison-Wesley (1989), which is incorporated by reference herein in its entirety. 
     Error Correction Controller 
     For closed form embodiments modeled as open kinematic chains, the error correction controller is identical to that presented for recursive computations in equation 26, with the exception that the subscript i has been removed, as described in equation 47 below.
 
 {umlaut over (q)}*=a{umlaut over (q)}   m   +K   p ( q   m   −q )+ K   v ( {dot over (q)}   m   −{dot over (q)} )  (47)
 
     Inverse Dynamics Module  420   
     For closed form embodiments modeled as open kinematic chains, the control law in equation 48 below is used estimate the joint loads that when applied to the system in equation 46 will reproduce the measured kinematics.
 
 U=M ( q ) {umlaut over (q)}*+H )( q,{dot over (q)} ) {dot over (q)}+G )( q )  (48)
 
     Forward Dynamics Module  425   
     For closed form embodiments modeled as open kinematic chains, the induced acceleration vector is computed using equation 49 below.
 
 {umlaut over (q)}=M   −1 ( q )( U−H ( q,{dot over (q)} ) {dot over (q)}−G ( q ))  (49)
 
     The closed loop error dynamics of the control law given by equation 48 is identical to those obtained in the recursive formulation. 
     B. Closed form Dynamics Embodiments With Closed Kinematic Chains 
     For closed form embodiments modeled as closed kinematic chains, such as the double support phase of motion, the effect of constraint forces is included in the equations of motion. The equation of motion for closed form embodiments modeled as closed kinematic chains is given by equation 50 below. In equation 50, J is the Jacobian matrix, and F is a constraint force vector.
 
 M ( q ) {umlaut over (q)}+H ( q,{dot over (q)} ) {dot over (q)}+G ( q )= U+J   T   F   (50)
 
     Inverse Dynamics Module  420   
     One problem with conventional techniques is that a unique solution to the system of equation 50 does not exist unless the constraint force F can be measured. One embodiment of the present invention provides a technique for determining a control law U by eliminating the term J T F from equation 50 by null-space projection. Elimination of constraint forces using the null-space projection has been previously used for bi-pedal control, as described in Jalics, Hemami, and Clymer, A Control Strategy for Terrain Adaptive Bipedal Locomotion, Autonomous Robots, 4: 243-257 (1997), which is incorporated by reference herein in its entirety. 
     Let N be the null-space of J T . Pre-multiplying equation 50 by N and incorporating the error correction feedback term {umlaut over (q)}* results in equation 51 below. The term NJ T F is zero because N is orthogonal to J T . The resulting system may be expressed in the form of equation 52 below. The term b is described in equation 53 below.
 
 N{M ( q ) {umlaut over (q)}*+H ( q,{dot over (q)} ) {dot over (q)}+G ( q )}= NU   (51)
 
NU=b  (52)
 
 b=N{M ( q ) {umlaut over (q)}*+H ( q,{dot over (q)} ) {dot over (q)}+G ( q )}  (53)
 
     Since N is not a square matrix in general, a pseudo-inverse solution that minimizes the energy cost function is given by equation 54 below. One skilled in the art will be able to obtain other cost functions that use more parameters than energy consumption to determine a movement profile.
 
 U=N   T ( NN   T ) −1   b   (54)
 
     In an alternative embodiment, joint loads U can be described by the product of a moment arm matrix and muscle forces. If muscles are used as actuators, determining the muscle forces may be achieved using a static optimization solution as described in the following literature, which is hereby incorporated by reference in its entirety: Crowninshield &amp; Brand, A physiologically based criterion of muscle force prediction in locomotion, Journal of Biomechanics, 14:793-801 (1981); Anderson &amp; Pandy, Static and Dynamic Solutions for Gait are Practically Equivalent, Journal of Biomechanics, 34:153-161 (2001), which are incorporated by reference herein in their entirety. 
     Forward Dynamics Module  425   
     For closed form embodiments modeled as closed kinematic chains, one way to compute the induced acceleration is described in equation 55 below.
 
 {umlaut over (q)}=M ( q ) −1 ( U+J   T   F−H ( q,{dot over (q)} ) {dot over (q)}−G ( q ))   (55)
 
     The forces of constraint F may be computed as functions of the state and the input U using equation 56 below. See Hemami &amp; Wyman, Modeling and Control of Constrained Dynamic Systems with Application to Biped Locomotion in the Frontal Plane, IEEE Transactions on Automatic Control, 24:526-535 (August 1979), which is incorporated by reference herein in its entirety.
 
 F =( JM ( q ) −1   J   T ) −1   {−{dot over (J)}{dot over (q)}+JM ( q ) −1 ( H ( q,{dot over (q)} ) {dot over (q)}+G ( q )− U )}  (56)
 
     Although the embodiment provided in equation 56 assumes a hard constraint, one skilled in the art will recognize that other models of contact, including penalty-based methods, can be used to compute the constraint forces. 
     IV. Synthesis of Novel Motion 
     In various embodiments of the present invention, forward dynamics module  420  can be used to predict novel motions. See Anderson &amp; Pandy, Dynamic Optimization of Human Walking, Journal of Biomechanical Engineering, 123:381-390, 2001, which is incorporated by reference herein in its entirety. As described above, inverse dynamics module  420  can be used to estimate joint loads necessary for a serial chain system to follow a desired trajectory. For example, inverse dynamics module  420  can calculate the motor commands necessary to drive a neuromuscular system based on desired trajectory information. Forward dynamics module  425  uses the estimated joint loads to calculate simulated kinematic data, which represents segmental position and velocity data for the serial chain system. 
     In various embodiments, forward dynamics module  425  may be used to predict the simulated kinematics of a serial chain system based on changing one or more original parameters to modified parameters. In various predictive embodiments, error correction controller  415  computes modified acceleration {umlaut over (q)}* using original kinematic data that is based on original parameters, and that represents a normal, learned, measured, or observed trajectory. Inverse dynamics module  420  estimates joint loads U based on the original parameters, and forward dynamics module  425  predicts simulated kinematics data based on modified parameters. 
     In one embodiment, forward dynamics module  425  may be used to predict the kinematic response resulting from changing segment parameters such as mass, inertia, length or center of mass. In another embodiment, forward dynamics module  425  may be used to predict the kinematic response resulting from surgical alterations, such as transferring a muscle from one insertion point to another. See Piazza &amp; Delp, Three-Dimensional Dynamic Simulation of Total Knee Replacement Motion During a Step-Up Task, Journal of Biomechanical Engineering, 123:589-606, 2001, which is incorporated by reference herein in its entirety. In yet another embodiment, forward dynamics module  425  may be used to predict the kinematic response resulting from loading the system. An example of loading the system is an external load in the form of a backpack attached to one of the segments. 
     According to one embodiment,  FIG. 6A  represents a tracking system in which forward dynamics module  425  is used to predict the simulated kinematics  610  for an open chain system with measured kinematics  450  available as input. The tracking system of the present invention does not require reaction forces and moments to be measured or predicted for an open chain system. According to another embodiment,  FIG. 6B  represents a tracking system in which forward dynamics module  425  is used to predict the simulated kinematics  610  for a closed chain system with measured kinematics  450  available as input. In one embodiment of the system in  FIG. 6B , reaction forces and moments  115  ( FIG. 1 ) are predicted for input into forward dynamics module  425 . Constraints module  615  uses equation  56  above to predict reaction forces  620  inputted into forward dynamics module  425 . Forward dynamics module  425  uses the predicted reaction forces  620  to computed induced acceleration {umlaut over (q)} as described in equation  55  above. Further, forward dynamics module  425  predicts the simulated kinematics  610  for the closed chain system by numerical integration of the induced acceleration {umlaut over (q)}. 
     According to one embodiment of the present invention, novel motion is predicted using an approach of recursive dynamics. According to another embodiment of the present invention in which a serial chain system is modeled as an open kinematic chain as described above with reference to  FIG. 6A , novel motion is predicted using an approach of closed form dynamics. 
     According to yet another embodiment of the present invention in which a serial chain system is modeled as a closed kinematic chain as described above with reference to  FIG. 6B , novel motion is predicted using an approach of closed form dynamics. To provide one example of this embodiment, reaction forces  620  are analytically estimated using joint loads  480  and simulated kinematics  610 , as described above with reference to equation  56 . To provide another example of this embodiment, joint loads  480  are determined using a null-space projection technique, as described above with reference to equations 51 through 54. 
     V. Simulation of an Open Chain System 
     Simulations conducted using measured data show the efficacy of the tracking system of an embodiment of the tracking system for estimating joint forces and moments. The simulated motion for experimentally obtained measured data is a cycle of human gait. 
       FIG. 7  is a free body diagram illustrating a four segment, three-dimensional lower extremity model that was used in the simulations. The four segments include pelvis segment  745 , thigh segment  750 , shank segment  755  and foot segment  760 . Hip joint  705  connects pelvis segment  745  and thigh segment  750 . Knee joint  710  connects thigh segment  750  and shank segment  755 . Ankle joint  715  connects shank segment  755  and foot segment  760 .  FIG. 7  depicts the joint loads at hip joint  705 , knee joint  710  and ankle joint  715 . Further,  FIG. 7  depicts segment parameters for thigh segment  750 , shank segment  755  and foot segment  760 . The variables used to represent joint loads and segment parameters have been described in detail with reference to  FIGS. 2 and 3  above. In particular, the ground reaction forces and moments, denoted by Γ gr  and N gr  correspond to the reaction forces and moments acting at the distal end of the last segment in the chain, i.e. at foot segment  760 . 
     A. Experimentally Measured Gait Data 
     Standard gait measurements including motion capture and force plate data were obtained from Vaughan, Davis and O&#39;Connor, Dynamics of Human Gait, Kiboho Publishers, Cape Town, South Africa, 2 nd  ed., 1999 (“Vaughan”), which is incorporated by reference in its entirety. The recorded motion from a Helen Hayes marker set and a series of anthropometric measurements were used to determine the body segment parameters and joint centers based on the statistical regression equations provided by Vaughan. The Euler angles and center of gravity of each body segment were calculated in order to construct the measured coordinates q m     i    for each segment. 
     While Euler angles are typically considered in the representation of relative rotational motion between articulating bodies, they do not provide anatomically meaningful description of joint motion. Likewise, joint reaction forces and moments do not convey anatomically meaningful information about the kinetics of motion. The method adopted by Vaughan for human gait analysis was used to display anatomically meaningful kinematics and kinetics. See Grood and Suntay, A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee, Journal of Biomechanical Engineering, 105:136-144, 1983, which is incorporated by reference herein in its entirety. The measured Euler angles Θ m  and simulated Euler angles Θ were converted to anatomically meaningful joint kinematics shown in  FIGS. 8 ,  9  and  10 . Similarly, the joint reaction moments N were converted to anatomically meaningful joint moments as shown in  FIGS. 11 ,  12  and  13 . 
     B. Tracking System Equations for the Lower Extremity System in FIG.  7   
     The equations for the tracking system  FIG. 4  are presented for the lower extremity system considered in  FIG. 7 . 
     Inverse Dynamics Module  420   
     According to one embodiment, inverse dynamics module  420  performs a recursive procedure based on equation 29. The equations required to obtain the force and moment balance estimates starting with joint loads U 4  at ankle  715 , followed by joint loads U 3  at knee  710 , and joint loads U 2  at hip  705  are described in equation 57 below.
 
 U   4   =A   p   −1 ( q   4 ){ M   4 ( q   4 ) {umlaut over (q)}   4   *−P   4 ( q   4   , {dot over (q)}   4 )− A   d     4   ( q   4 ) U   gr }
 
 U   3   =A   p   −1 ( q   3 ){ M   3 ( q   3 ) {umlaut over (q)}   3   *−P   3 ( q   3   , {dot over (q)}   3 )− A   d     3   ( q   3 ) U   4 }
 
 U   2   =A   p   −1 ( q   2 ){ M   2 ( q   2 ) {umlaut over (q)}   2   *−P   2 ( q   2   , {dot over (q)}   2 )− A   d     2   ( q   2 ) U   3 }  (57)
 
     The output of inverse dynamics module  420  is a vector containing the forces and moments at all three joints, represented in equations 58 below.
 
U=[U 2   T U   3   T U   4   T ] T   (59)
 
     Error Correction Controller  415   
     From equation 26 above, equations for error correction controller  415  are given by equations 59, 60 and 61 below.
 
 {umlaut over (q)}   4   *=a{umlaut over (q)}   m     4     +K   p     4   ( q   m     4     −q   4 )+ K   v     4   ( {dot over (q)}   m     4     −{dot over (q)}   4 )  (59)
 
 {umlaut over (q)}   3   *=a{umlaut over (q)}   m     3     +K   p     3   ( q   m     3     −q   3 )+ K   v     3   ( {dot over (q)}   m     3     −{dot over (q)}   3 )  (60)
 
 {umlaut over (q)}   2   *=a{umlaut over (q)}   m     2     +K   p     2   ( q   m     2     −q   2 )+ K   v     2   ( {dot over (q)}   m     2     −{dot over (q)}   2 )  (61)
 
     The output of error correction controller  415  is an array represented in equation 62 below.
 
{umlaut over (q)}*=[{umlaut over (q)} 2 * T  {umlaut over (q)} 3 * T  {umlaut over (q)} 4 * T ] T   (62)
 
     Forward Dynamics Module  425   
     The induced linear and angular acceleration vector for each segment is computed from equation 31 as described in equations 63 below. The induced acceleration array {umlaut over (q)}* for all three segments is given by equation 64 below.
 
 {umlaut over (q)}   4   =M   4   −1 ( q   4 )[ P   4 ( q   4   , {dot over (q)}   4 )+ A   p     4   ( q   4 ) U   4   +A   p     4   ( q   4 ) U   gr ]
 
 {umlaut over (q)}   3   =M   3   −1 ( q   3 )[ P   3 ( q   3   , {dot over (q)}   3 )+ A   p     3   ( q   3 ) U   3   +A   d     3   ( q   3 ) U   4 ]
 
 {umlaut over (q)}   2   =M   2   −1 ( q   2 )[ P   2 ( q   2   , {dot over (q)}   2 )+ A   p     2   ( q   2 ) U   2   +A   d     2   ( q   2 ) U   3 ]  (63)
 
{umlaut over (q)}=[q 2   T  q 3   T  q 4   T ] T   (64)
 
     Using well-known techniques of numerical integrations, forward dynamics module  425  numerically integrates induced acceleration array {umlaut over (q)}* to obtain the vectors q and {dot over (q)}, which are represented in equations 65 below. According to one embodiment, the numerical integration method used for the purpose of the present simulations is a first order Euler method.
 
{dot over (q)}=[{dot over (q)} 2   T  {dot over (q)} 3   T  {dot over (q)} 4   T ] T  
 
q=[q 2   T  q 3   T  q 4   T  ] T   (65)
 
     The individual matrices in the above equations of motion are described for foot segment  760 , shank segment  755  and thigh segment  750  in equations 66, 67 and 68 below. 
     Foot Segment  760   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             M 
                             4 
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               4 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     m 
                                     4 
                                   
                                   ⁢ 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 ∅ 
                               
                               
                                 
                                   
                                     I 
                                     4 
                                   
                                   ⁢ 
                                   
                                     H 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             P 
                             4 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 q 
                                 4 
                               
                               , 
                               
                                 
                                   q 
                                   . 
                                 
                                 4 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                     
                                 
                               
                               
                                 
                                   
                                       
                                   
                                   - 
                                 
                               
                               
                                 
                                   
                                     
                                       m 
                                       4 
                                     
                                     ⁢ 
                                     g 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                               
                                 
                                     
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     4 
                                   
                                 
                               
                               
                                 
                                   
                                     H 
                                     . 
                                   
                                   4 
                                 
                               
                               
                                 
                                   
                                     q 
                                     . 
                                   
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     f 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         W 
                                         4 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               d 
                               4 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               4 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       R 
                                       o 
                                         
                                         
                                         
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       4 
                                       , 
                                       cop 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               p 
                               4 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               4 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   I 
                                   _ 
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     4 
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       4 
                                       , 
                                       4 
                                     
                                   
                                 
                               
                               
                                 
                                   R 
                                   o 
                                     
                                     
                                     
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           4 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   4 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   4 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           gr 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   gr 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   gr 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     Shank Segment  755   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             M 
                             3 
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               3 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     m 
                                     3 
                                   
                                   ⁢ 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 ∅ 
                               
                               
                                 
                                   
                                     I 
                                     3 
                                   
                                   ⁢ 
                                   
                                     H 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             P 
                             3 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 q 
                                 3 
                               
                               , 
                               
                                 
                                   q 
                                   . 
                                 
                                 3 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                     
                                 
                               
                               
                                 
                                   
                                       
                                   
                                   - 
                                 
                               
                               
                                 
                                   
                                     
                                       m 
                                       3 
                                     
                                     ⁢ 
                                     g 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                               
                                 
                                     
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     
                                       I 
                                       - 
                                     
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     H 
                                     . 
                                   
                                   3 
                                 
                               
                               
                                 
                                   
                                     q 
                                     . 
                                   
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     f 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         W 
                                         3 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               d 
                               3 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               3 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       R 
                                       o 
                                         
                                         
                                         
                                       3 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       3 
                                       , 
                                       4 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               p 
                               3 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               3 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   I 
                                   _ 
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     3 
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       3 
                                       , 
                                       3 
                                     
                                   
                                 
                               
                               
                                 
                                   R 
                                   o 
                                     
                                     
                                     
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           3 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   3 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   3 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           4 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   4 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   4 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     Thigh Segment  750   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             M 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               2 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     m 
                                     2 
                                   
                                   ⁢ 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 ∅ 
                               
                               
                                 
                                   
                                     I 
                                     2 
                                   
                                   ⁢ 
                                   
                                     H 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             P 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 q 
                                 2 
                               
                               , 
                               
                                 
                                   q 
                                   . 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                     
                                 
                               
                               
                                 
                                   
                                       
                                   
                                   - 
                                 
                               
                               
                                 
                                   
                                     
                                       m 
                                       2 
                                     
                                     ⁢ 
                                     g 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                               
                                 
                                     
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     H 
                                     . 
                                   
                                   2 
                                 
                               
                               
                                 
                                   
                                     q 
                                     . 
                                   
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     f 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         W 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               d 
                               2 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               2 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   
                                     I 
                                     _ 
                                   
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       R 
                                       o 
                                         
                                         
                                         
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       2 
                                       , 
                                       3 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             
                               p 
                               2 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               2 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   I 
                                   _ 
                                 
                               
                               
                                 ∅ 
                               
                             
                             
                               
                                 
                                   
                                     R 
                                     o 
                                       
                                       
                                       
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       ~ 
                                     
                                     
                                       2 
                                       , 
                                       2 
                                     
                                   
                                 
                               
                               
                                 
                                   R 
                                   o 
                                     
                                     
                                     
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           2 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   2 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   2 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           U 
                           4 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   Γ 
                                   3 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   3 
                                     
                                     
                                     
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     C. Results of Simulations 
       FIGS. 8 ,  9 , and  10  compare simulated joint angle trajectories versus measured joint angle trajectories for a complete cycle of gait according to one embodiment. In these simulations, the position feedback gain of K p =4000 was used and the acceleration terms {umlaut over (q)} m  were omitted in the feedback structure given by Equation 26, i.e. a=0. It is observed that the tracking performance of tracking controller  410  is exceptional in spite of the fact that estimated accelerations {umlaut over (q)} m  were ignored. Various embodiments can further improve the tracking performance by decreasing the sampling rate for numerical integration in forward dynamics module  425  and/or by increasing feedback gains. One skilled in the art will recognize that decreasing the sampling rate requires re-sampling the original signals at a higher frequency. One way to improve tracking performance is by using cubic splines to re-sample the measured kinematic data at a higher frequency. It has been observed that first order numerical integration methods such as Euler&#39;s method yield suitable results. 
       FIGS. 11 ,  12 , and  13  compare the joint moments at ankle joint  715 , knee joint  710 , and hip joint  705  respectively using traditional inverse dynamics analysis to those obtained using the forward dynamics solution of the present invention. It is observed that if the tracking error is minimized, then the forward dynamics solution will approach the inverse dynamics solution. The forward dynamics solution advantageously avoids the use of accelerations {umlaut over (q)} m  estimated from measured kinematics data. Therefore, in presence of noise, the forward dynamics solution of the present invention will outperform the inverse dynamics solution. 
     The present invention may be embodied in various forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that disclosure will be thorough and complete and will fully convey the invention to those skilled in the art. Further, the apparatus and methods described are not limited to rigid bodies. 
     While particular embodiments and applications of the present invention have been illustrated and described herein, it is to be understood that the invention is not limited to the precise construction and components disclosed herein and that various modifications, changes, and variations may be made in the arrangement, operation, and details of the methods and apparatuses of the present invention without department from the spirit and scope of the invention as it is defined in the appended claims.