Patent Publication Number: US-11048275-B1

Title: Enhanced guidance laws for precision operations

Description:
BACKGROUND 
     Controlled guidance of a mobile platform such as a vehicle requires controllers with control algorithms that are precise, dependable, and robust. Although some conventional algorithms may enable the mobile platform to go from point ‘A’ to point ‘B’ safely under ideal circumstances, traversing large distances usually results less than ideal circumstances. As the use of artificial intelligence (AI) and abilities of self-controlled (e.g., autonomous) vehicles increases, the precision, dependability, and robustness of control algorithms incorporated in these vehicles correspondingly needs to increase to successfully deal with less than ideal circumstances. 
     Often multiple a priori parameters and constraints are computed to determine a defined path for a mobile platform. During real-time operation, these parameters and constraints are input into the mobile platform to enable real-time tracking of a desired path, the desired path representing an approximation of an ability of the mobile platform to track the defined path. The ability of the mobile platform to track the defined path or to maintain the desired path is directly related to the control algorithms incorporated in guidance control systems and their response to disturbances. 
     Another factor of consideration in traversing large distances is fuel economy. Control algorithms that are not able to maintain strict tolerances often result unnecessary fuel expenditures. Further, with respect to manned vehicles, operators often do not recognize when strict tolerances are not being maintained. 
     For passenger vehicles, sudden or rapid fluctuations in response to control commands increases passenger discomfort and peace of mind. These sudden or rapid fluctuations may be due to inadequate error reduction or error compensation in control algorithms, unfiltered control parameters, improper reference points or reference frames, overcompensating disturbance variables, improper consideration of a response delay, or combinations thereof. 
     Therefore, it is desirous to obtain more robust, secure, and otherwise improved guidance algorithms, systems, methods, and apparatuses. 
     SUMMARY 
     In one aspect, the inventive concepts disclosed herein are directed to a system for precision guidance incorporating enhanced guidance laws. The system may include a platform, one or more sensors on-board the platform, and a navigational guidance computer on-board the platform. The navigational guidance computer may be configured for receiving one or more inputs. For example, the one or more inputs may include navigation data indicative of a defined path from a navigation database (NDB) and position data from the one or more sensors. The navigational guidance computer may be configured for comparing juxtaposed portions of the defined path to determine a discontinuity in the defined path. The navigational guidance computer may be further configured for defining a containment region and a curved reference line of a non-linear desired path that tracks the defined path. In this regard, the containment region may be defined according to a threshold distance on either side of the defined path, and the curved reference line may be interposed between the juxtaposed portions to span the discontinuity. The navigational guidance computer may be further configured for comparing a current platform position derived from the position data to a planned platform position derived from the navigation data to determine a platform deviation relative to the defined path, the desired path, and the curved reference line. The navigational guidance computer may be further configured for outputting a reference signal for controlling the platform relative to the defined path, the desired path, and the curved reference line, to compensate for the deviation and maintain the platform deviation within the threshold distance. 
     In a further aspect, the inventive concepts disclosed herein are directed to an apparatus for precision guidance. The apparatus may include one or more ports for receiving a first input from a database or data structure, receiving a second input from a sensor, and for outputting a reference signal for guiding or controlling a platform relative to a non-linear portion of a desired path. The apparatus may further include a processor configured to access a non-transitory memory with instructions thereon. The processor and the non-transitory memory with instructions may be configured for: processing the first input to compare juxtaposed portions of the defined path to determine a discontinuity in the defined path; defining a containment region and a curved reference line of a non-linear desired path that tracks the defined path, the containment region defined according to a threshold distance on either side of the defined path, and the curved reference line interposed between the juxtaposed portions to span the discontinuity; comparing a current platform position derived from the second input to a planned platform position derived from the first input to determine a platform deviation relative to the defined path, the desired path, and the curved reference line; and outputting the reference signal for controlling the platform relative to the defined path, the desired path, and the curved reference line, to compensate for the deviation and maintain the platform deviation within the threshold distance. 
     In a further aspect, the inventive concepts disclosed herein are directed to a method for precision guidance. The method may include receiving, via a processor and a memory of a platform, one or more inputs. For example, the one or more inputs may include navigation data indicative of a defined path from a navigation database (NDB) and position data from at least one sensor on-board the platform. The method may further include comparing, via the processor, juxtaposed portions of the defined path to determine a discontinuity in the defined path; defining, via the processor, a containment region and a curved reference line of a non-linear desired path that tracks the defined path, the containment region defined according to a threshold distance on either side of the defined path, and the curved reference line interposed between the juxtaposed portions to span the discontinuity; comparing, via the processor, a current platform position derived from the position data to a planned platform position derived from the navigation data to determine a platform deviation relative to the defined path, the desired path, and the curved reference line; and outputting, to a control system, a reference signal for controlling the platform relative to the defined path, the desired path, and the curved reference line, to compensate for the deviation and maintain the platform deviation within the threshold distance. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Implementations of the inventive concepts disclosed herein may be better understood when consideration is given to the following detailed description thereof. Such description makes reference to the included drawings, which are not necessarily to scale, and in which some features may be exaggerated and some features may be omitted or may be represented schematically in the interest of clarity. Like reference numerals in the drawings may represent and refer to the same or similar element, feature, or function. In the drawings: 
         FIG. 1  is an exemplary embodiment of systems with enhanced guidance laws, according to the inventive concepts disclosed herein; 
         FIG. 2  is an exemplary embodiment of a system with enhanced guidance laws, according to the inventive concepts disclosed herein; 
         FIG. 3  is an exemplary embodiment of an alert system, according to the inventive concepts disclosed herein; 
         FIG. 4  is an exemplary embodiment of a system with enhanced guidance laws, according to the inventive concepts disclosed herein; 
         FIG. 5  is an exemplary embodiment of a block diagram indicating flow of one or more signals with respect to a guidance module, according to the inventive concepts disclosed herein; 
         FIG. 6  is an exemplary embodiment of one or more controller configurations, according to the inventive concepts disclosed herein; 
         FIG. 7  is an exemplary embodiment of lateral curve, according to the inventive concepts disclosed herein; 
         FIG. 7A  is an exemplary embodiment of a control deviation and an integrity deviation, according to the inventive concepts disclosed herein; 
         FIG. 8  is an exemplary embodiment of a coordinated turn using enhanced guidance laws and force vectors associated with a platform during the turn, according to the inventive concepts disclosed herein; 
         FIG. 9  is an exemplary embodiment of a vector diagram for determining a control command, according to the inventive concepts disclosed herein; 
         FIGS. 10A and 10B  are exemplary embodiments of filters for filtering one or more signals, according to the inventive concepts disclosed herein; 
         FIG. 11  is an exemplary embodiment of a vertical curve, according to the inventive concepts disclosed herein; 
         FIG. 12  is an exemplary embodiment of controller modes with respect to the defined path, according to the inventive concepts disclosed herein; 
         FIG. 13  is an exemplary embodiment of an arc connecting legs of the defined path, according to the inventive concepts disclosed herein; 
         FIG. 14  is an exemplary embodiment of a method for navigational guidance using enhanced guidance laws, according to the inventive concepts disclosed herein; 
         FIG. 15  is an exemplary embodiment of sub-steps of the method of  FIG. 14 , according to the inventive concepts disclosed herein; 
         FIG. 16  is an exemplary embodiment of a system with enhanced guidance laws, according to the inventive concepts disclosed herein; 
         FIG. 17  is an exemplary embodiment of an approach plan, according to the inventive concepts disclosed herein; 
         FIG. 18  is an exemplary embodiment of a chart depicting deviations being maintained within a lateral containment region, according to the inventive concepts disclosed herein; 
         FIG. 19  is an exemplary embodiment of a chart depicting deviations being maintained within a vertical containment region, according to the inventive concepts disclosed herein; 
         FIG. 20  is an exemplary embodiment of a chart depicting performance of a roll command, according to the inventive concepts disclosed herein; 
         FIG. 21  is an exemplary embodiment of a chart depicting performance of an aircraft roll, according to the inventive concepts disclosed herein; 
         FIG. 22  is an exemplary embodiment of a diagram depicting one or more portions of an approach, according to the inventive concepts disclosed herein; 
         FIG. 23  is an exemplary embodiment of a plot of a cross-track velocity, a filtered cross-track velocity, and a cross-track velocity determined using a derivative function, according to the inventive concepts disclosed herein; 
         FIG. 24  is an exemplary embodiment of a chart depicting a performance of a complimentary filter, according to the inventive concepts disclosed herein; 
         FIG. 25  is a chart depicting aircraft performance, according to one or more conventional systems; 
         FIG. 26  is a chart depicting control deviations and integrity deviations, according to one or more conventional systems; 
         FIG. 27  is an exemplary embodiment of a chart depicting aircraft performance, according to the inventive concepts disclosed herein; 
         FIG. 28  is an exemplary embodiment of a chart depicting lateral deviations from a defined path, according to the inventive concepts disclosed herein; 
         FIG. 29  is an exemplary embodiment of a chart depicting roll anticipation time versus delta bank angle required, according to the inventive concepts disclosed herein; 
         FIG. 30  is an exemplary embodiment of multiple charts depicting aircraft performance with respect to a Capture Mode, according to the inventive concepts disclosed herein; 
         FIG. 31  is an exemplary embodiment of a chart depicting aircraft performance with respect to a pitch command, according to the inventive concepts disclosed herein; and 
         FIGS. 32-35  are exemplary embodiments of charts depicting aircraft performance, according to the inventive concepts disclosed herein. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Before explaining at least one embodiment of the inventive concepts disclosed herein in detail, it is to be understood that the inventive concepts are not limited in their application to the details of construction and the arrangement of the components or steps or methodologies set forth in the following description or illustrated in the drawings. In the following detailed description of embodiments of the instant inventive concepts, numerous specific details are set forth in order to provide a more thorough understanding of the inventive concepts. However, it will be apparent to one of ordinary skill in the art having the benefit of the instant disclosure that the inventive concepts disclosed herein may be practiced without these specific details. In other instances, well-known features may not be described in detail to avoid unnecessarily complicating the instant disclosure. The inventive concepts disclosed herein are capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting. 
     As used herein a letter following a reference numeral is intended to reference an embodiment of the feature or element that may be similar, but not necessarily identical, to a previously described element or feature bearing the same reference numeral (e.g., 1, 1a, 1b). Such shorthand notations are used for purposes of convenience only, and should not be construed to limit the inventive concepts disclosed herein in any way unless expressly stated to the contrary. 
     Further, unless expressly stated to the contrary, “or” refers to an inclusive or and not to an exclusive or. For example, a condition A or B is satisfied by anyone of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), and both A and B are true (or present). 
     In addition, use of the “a” or “an” are employed to describe elements and components of embodiments of the instant inventive concepts. This is done merely for convenience and to give a general sense of the inventive concepts, and “a” and “an” are intended to include one or at least one and the singular also includes the plural unless it is obvious that it is meant otherwise. 
     Finally, as used herein any reference to “one embodiment,” or “some embodiments” means that a particular element, feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the inventive concepts disclosed herein. The appearances of the phrase “in some embodiments” in various places in the specification are not necessarily all referring to the same embodiment, and embodiments of the inventive concepts disclosed may include one or more of the features expressly described or inherently present herein, or any combination of sub-combination of two or more such features, along with any other features which may not necessarily be expressly described or inherently present in the instant disclosure. 
     “Guidance laws” as used herein shall mean control algorithms including feedback control algorithms useful in controllers (e.g., proportional-integral (PI) controller, proportional-derivative (PD) controllers, or proportional-integral-derivative (PID) controllers) to effect a beneficial navigational outcome based on controller gain, one or more controlled variables, or a set-point, despite effects of one or more disturbance variables (e.g., wind, or other external forces). In feedback control, the goal is to reduce an error value or error signal associated with a control signal. 
     “Decision Altitude” (DA) as used herein shall mean an altitude at which a missed approach must be initiated if required references to continue the approach are not established. In some embodiments, DA is used in the context of a platform making an approach using vertical guidance capabilities. 
     “Maneuver” as used herein shall mean the use of kinematic and potential forces to produce a movement or a series of movements, which are usually performed by an object, machine, or vehicle, with respect to a reference point or destination (e.g., performing final approach segment of a flight path). 
     “Area Navigation System” (RNAV) a navigation system permitting an aircraft to operate in or on any desired flight path within coverage of ground/spaced-based navigation aids or within the limits, capacity, or capability of self-contained aids, or combinations thereof. 
     “Required Navigational Performance” (RNP) as used herein shall mean a subset of RNAV. The RNP system or method meets a performance requirement of 95% probability that the navigation system accuracy remains within a limit or degree of accuracy defined for the RNAV or RNP operation. For example, during an RNP-0.1 operation, the total system error (TSE) remains within 0.1 nautical mile (NM) of a desired path 95% of the time, and an on-board monitoring system provides an alert to the operator when TSE exceeds the probability threshold. By way of another example, during an RNP-1.0 operation, a pilot may be provided with an alert when the probability that TSE exceeds 2 NM is greater than 10 −5 . RNP procedures apply to, but are not limited to, instrument departure procedures, standard terminal arrivals, and instrument approaches. It is noted that RNP accuracy may vary with a segment of a defined path performed or tracked in real-time. For example, an accuracy of RNP-1.0 may be applicable in an Initial, Intermediate, or Missed Approach segment, while an accuracy of RNP-0.3 may be applicable in a Final Approach Segment. 
     “Memory” as used herein, and unless otherwise specified, means any storage medium known in the art suitable for storing program instructions executable by an associated one or more processors (e.g., computer executable program code). Memory may also be stored in an organized format, encompassing, for example, a database. Memory may include one or more storage mediums. For example, memory may include, but is not limited to, a read-only memory, a random access memory, a magnetic or optical memory device (e.g., disk), a magnetic tape, a solid state drive, or combinations thereof. In embodiments, memory includes a buffer (e.g., frame buffer) and/or a cache. The memory includes non-transitory memory. In embodiments, the memory may be located remotely with respect to the platform. 
     “Processor” as used herein means any processing device, including but not limited to, a microprocessor, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a central processing unit (CPU), an arithmetic logic unit (ALU), a digital signal processor (DSP), or combinations thereof. 
     “Module,” “block” or “sub-block” as used herein means a combination of hardware and software configured to perform one or more steps, processes and/or algorithms of the inventive concepts disclosed herein. 
     “Flight Management System” (FMS) as used herein shall mean a system having a sensor, receiver, and processor with navigation and aircraft performance databases integrated to perform, manage, direct, or control RNAV guidance to a flight display and automatic flight control system. 
     “Flight Technical Error” (FTE) as used herein shall mean an individual component of TSE. The FTE of an aircraft dictates an accuracy to which the aircraft is controlled according to a function or relationship of aircraft position relative to a control output, command, or a desired position. 
     “Lateral Navigation” (LNAV) as used herein shall mean a function of an RNAV or RNP system directly responsible for computed, displayed, managed, controlled, performed, or otherwise provided lateral guidance to track a desired path. 
     “Vertical Navigation” (VNAV) as used herein shall mean a function similar to LNAV except that it is with respect to vertical guidance. 
     “Path Definition Error” (PDE) as used herein shall mean a difference between a defined path and a desired path. 
     “Leg” as used herein shall mean a portion of a total traversed path, such as a flight path. The term includes, but is not limited to, a geodesic line between two fixes (TF leg), a direct path to a fix (DF leg), a specified track to a fix that is defined by a course (CF leg), a specified track to an altitude (FA leg), and an RF leg (below). 
     “Radius to a Fix (RF) Leg” as used herein shall mean a constant radius circular path around a defined reference point (e.g., turn center) that starts and terminates at a fix. As used herein, this term encompasses both lateral turns and curved descents. In an exemplary embodiment, the constant radius circular path is a constant circular ground path. 
     Broadly, the inventive concepts disclosed herein are directed to systems, methods, apparatuses, and algorithms with improved or enhanced guidance laws as compared to conventional systems, methods, apparatuses, or algorithms. The enhanced guidance laws provide abilities to a mobile platform to accurately track a desired path, including a non-linear path, despite non-ideal conditions. Warnings or alerts are provided when the tracking is inaccurate. Reference frames for computing controller gain, set-points, or tuning parameters more precisely represent real-time scenarios by relying on earth-surface (e.g., ground) reference points, as opposed to relative air masses. Further, transitions between modes of a multi-mode controller and discontinuities between legs of the desired path are smoothed by extending a period of time associated with the transitions or by reducing deviations associated with the discontinuities. The enhancements to the guidance laws are verified by examples including statistical performance tests. 
     Referring now to  FIG. 1 , embodiments of the inventive concepts disclosed herein are directed to a mobile platform  102   a  and/or  102   b  incorporating enhanced guidance laws to track a desired path  104   a  and/or  104   b . The desired path  104   a  and/or  104   b  including one or more non-linear or curved portions, which are tracked while accounting for external forces (e.g., wind) and other non-idealities. For example, the mobile platform  102   a  or  102   b  may include a vehicle such as a manned aircraft, unmanned aircraft such as a UAV or UAS, or a submarine that traverses a fluidic medium such as air or water, while following the desired path  104  using LNAV and/or VNAV of one or more control systems. In this regard, the desired path  104  may be derived from one or more thresholds  106  (e.g.,  FIGS. 7, and 11 , below) and a defined path  108  (e.g.,  FIG. 7 , and  FIG. 11 , below) minus any PDE. The defined path  108  may be a result of a priori data received from one or more memories associated with the mobile platform  102 . 
     In an exemplary embodiment, the desired path  104  may be a curved ground track, a first curvature of which is attributable to a curvature of the Earth. For example, the first curvature of the desired path  104   a  may be approximated by an ellipsoidal Earth model. By way of another example, the first curvature may be approximated by a spherical Earth model. In some embodiments, the first curvature may be approximated by both an ellipsoidal Earth model and a spherical Earth model, such as determining the first curvature based on a weighted or averaged solution based on results from both models. 
     In other embodiments, a second curvature of the desired path  104  may be due to a descent, an ascent, or a turn segment in the defined path  108  (below) being tracked by the desired path  104 . For example, the second curvature of the desired path  104   b  may be due to a descent in the defined path  108   b . In some embodiments, the second curvature of the desired path  104  may be due to both a turn segment and a descent. 
     Referring now to  FIG. 2 , an exemplary embodiment of a system  100  incorporating enhanced guidance laws is depicted. The system  100  may include the mobile platform  102 , a processor  110 , a memory  112 , a sensor  114 , a control system  116 , and an alert system  118 . For example, the processor  110  may include one or more processors integrated to form a navigational guidance computer configured to receive navigational data (e.g., the a priori data and/or updated information) from the memory  112 . It is noted that some components of system  100  may be located on-board the platform  102 , while others may be located off of the platform  102 , but at least the processor  110  and a portion of the memory  112  is located on-board the platform  102 . 
     In an exemplary embodiment, the processor  110  may be communicatively coupled to the memory  112  and the sensors  114  to receive the defined path  108  and position data, including a position of the platform  102  relative to a desired path  104 . 
     In some embodiments, the processor  110  may include multiple processors such as multiple CPUs connected by a central interconnect (CI)  111 . For example, the CI  111  may be a shared system memory such as shared Random Access Memory (RAM). By way of another example, the CI  111  may include a peripheral component interconnect (PCI), PCI Express (PCIe), or PCIe Fiber input/output (I/O) board, expansion module, interface, or combinations thereof. In other embodiments, the processor  110  may include multiple cores, with a first one or more cores of the multiple cores (e.g., off-load engine) dedicated to processing using one or more commercial communication protocols (e.g., Ethernet), and a second one or more of the multiple cores dedicated to processing using one or more non-commercial or non-Ethernet protocols (e.g., ARINC 429 or MIL-STD-1553). 
     In an exemplary embodiment, the enhanced guidance laws disclosed herein, or functions of the enhanced guidance laws, are performed at a rate that is proportional to one or more processors of the processor  110 . For example, if a guidance law function is run at a rate of 10 Hz, then a time associated with performing the function may be 0.1 seconds. By way of another example, if the guidance law function is run at a rate of 40 MHz, then a time associated with performing the function may be 4*10 −7 , or at 60 MHz, then the time associated with performing the function may be 6*10 −7  seconds, and so on and so forth. 
     In an exemplary embodiment, the memory  112  may include, but is not limited to, one or more memories functioning as a navigational database (NDB) to provide information including, but not limited to, path constraints, required times of arrival (RTAs), associated speed, altitude, waypoint, flight modes, uplinked wind data, temperature data, weather data, platform specifications (e.g., engine, lift, drag, age, performance capabilities, etc.), or combinations thereof. In some embodiments, the NDB provides navigational data in compliance with one or more standards to the navigational guidance computer. For example, a waypoint resolution error may be less than or equal to 60 feet (about 19 m), and a flight path angle may be stored in the NDB to a resolution of hundredths of a degree (0.01°), such that the defined path  108  may be within at least five feet (1.5 m) of a published path. Thus, in some embodiments, the NDB may be incompliance with AC 20-153 or an equivalent industry standard. 
     In an exemplary embodiment, the sensor  114  may include multiple sensors, including at least one on-board sensor that is located on the mobile platform  102 . The multiple sensors may include, but are not limited to, a global positioning system (GPS), a differential GPS, distance measuring equipment (DME), an inertial reference unit (IRU), a Light Detection and Ranging (LIDAR) sensor, a barometric altimeter, or combinations thereof. In some embodiments, a second sensor may be redundant equipment to a first sensor for use in a segment of flight such as a missed approach segment (MAS) to ensure compliance with an RNP value of RNP-1.0 or less. 
     In an exemplary embodiment, the control system  116  is configured as a multi-mode controller. For example, the multi-mode controller may be a multi-mode FMS, with the multiple modes including, but not limited to, an Altitude Hold Mode, a Capture Mode, a Transition to Track Mode, and a Track Mode. In other embodiments, the control system  116  is configured as a single mode controller. For example, the control system  116  may be implemented as a Lateral Path Mode (LPATH) controller or as a Vertical Path Mode (VPATH) controller. 
     In some embodiments, the control system  116  may be physically separated from other components of system  100 . For example, the control system  116  may include an autopilot, which may be implemented on-board the platform  102  and may be physically separated from a portion of the memory  112 . For instance, processor  110  may be implemented as a component of a secure UAS kit, where the mission computer storing some or all of the navigational database (NDB) is located at a ground station (not shown), while the autopilot of control system  116  is configured to be installed, attached, or otherwise located on the platform  102 . 
     In some embodiments, the physically separated components of system  100  may be communicatively coupled via one or more data links. For example, the one or more data links may provide point-to-point (P2P) communication and/or fully networked communication (e.g., Ethernet). For instance, the data link may include, but is not limited to, a Link16 signal, a Link2000 signal, or signals from Control and Non-Payload Communications (CNPC) such as a signal having a CNPC-1000 waveform. 
     In an exemplary embodiment, the alert system  118  may be communicatively coupled to the processor  110  to provide an alert to the operator when one or more thresholds  106  are not being maintained. For example, a first threshold  106 ( 1 ) (e.g.,  FIG. 7 , below) may be an outer bound and a second threshold  106 ( 2 ) may be an inner bound of the desired path  104  (e.g., inner and outer with respect to a turn center or an arc associated with the turn center), and the alert system  118  may include an audio system, a display, a user interface, or a combination thereof, configured to provide a tone, a vibration, a warning or alert signal, a speech announcement, or a combination thereof, when either the first threshold  106 ( 1 ) or the second threshold  106 ( 2 ) is not maintained relative to the desired path  104  and/or the defined path  108 . For instance, the defined path  108  may be a flight path and the desired path  104  may be defined by outer and inner bounds established by one or more RNP values such that a course  120  (e.g.,  FIG. 7 , below) an aircraft travels to track the defined path  108  remains within the inner and outer bounds of the desired path  104  as ensured by the alerts provided by the alert system  118 . 
     It is noted that in some embodiments, the course  120  may differ from the flight path  108  at least due to PDE associated with the desired path  104 . In other embodiments, the course  120  may differ from the flight path due to PDE, path steering error (PSE), and position estimation error (PEE) (e.g., difference between true position and estimated position), or combinations thereof. It is further noted that PDE combined with PSE and PEE may make up the TSE. 
     In an exemplary embodiment, the one or more thresholds  106  may be associated with one or more degrees of accuracy  107 . For example, for a threshold associated with the LNAV function, the inner bound may be a first distance ranging from 0.1 to 1.0 NM, or a second distance ranging from 1.0 NM to 10 NM, and the one or more degrees of accuracy may be associated with a probability of maintaining one or more of the first distance and the second distance relative to a reference point or the defined path  108  over time. For instance, when the probability that the TSE associated with the platform  102  is not within a 2×RNP value or will not be within the 2×RNP value at least 99.999% of the time, an alert may be provided. In other words, an alert may be provided if the probability of TSE being maintained within the threshold will not be in compliance with greater than or equal to a four sigma standard at a specified RNP value. 
     Referring now to  FIG. 3 , an exemplary embodiment of a display  122  with a visual alert system  118  is depicted. For example, the display  122  may include one or more indicators  118   a  and/or  118   b , which provide a visual notification to a pilot that either a vertical threshold distance associated with the indicator  118   a  or a lateral threshold distance associated with the indicator  118   b  is not being maintained by a current platform position. In other words, the indicator  118   a  or  118   b  indicates when the current platform position is approaching and/or exceeding, respectively, the vertical or the lateral threshold distance. In some embodiments, the visual indications may be accompanied by audio (e.g., tone), or sensory (e.g., vibrational) indications. 
     In an exemplary embodiment, a scale factor associated with the indicators  118   a  and  118   b  may adjust according to a portion of the defined path  108  being traversed. For example, a measurement scale (e.g., two circles, diamonds, hash marks, tick marks, dots, etc.) in the indicator  118   a  may coincide with 150 ft. (46 m) when starting a missed approach during the Final Approach (FA) segment of a flight path, and on that measurement scale a single vertical deviation may be 75 ft. (23 m). By way of another example, in a second measurement scale, a single vertical deviation may coincide with a 500 ft. (153 m) deviation. By way of another example, a third measurement scale for the indicator  118   b  may coincide with 1 NM. For instance, when starting a missed approach segment, the measurement scale for the indicator  118   b  may coincide with an RNP requirement of RNP-1.0, where a lateral deviation may be 1 NM; whereas during level flight, a fourth measurement scale may coincide with 2×RNP-1.0, where a lateral deviation may be 2 NM. In some embodiments, the display is configured to automatically adjust the measurement scale based on the approach or segment of flight. 
     Referring now to  FIG. 4 , an exemplary embodiment of a system  100   c  with enhanced guidance laws is depicted. The system  100   c  may function similarly to system  100 , and vice versa, except that portions of system  100   c  may be implemented as a single unit, an apparatus, or as one or more individual partitions. For example, the system  100   c  may include one or more processors  110   c , a memory  112   c , one or more sensors  114   c  (e.g., GPS, air data inertial reference unit (ADIRU), etc.), and a vehicle control system  116   c  (e.g., autopilot, digital engine control, etc.), where at least the processors  110   c  and a portion of the memory  112   c  may be implemented as a single unit  124 . For instance, the single unit  124  may include an application specific integrated circuit (ASIC), a Field Programmable Gate Array (FPGA), a system on a chip (SOC), a digital signal processor (DSP), or combinations thereof. 
     In some embodiments different components of the system  100   c  may be separated into one or more individual partitions. For example, at least the processors  110   c  and a portion of the memory  112   c  may be implemented on an individual partition (e.g.,  116 ), of multiple partitions of the system  100   c  such that logic, memory, or information for each individual partition may be managed and/or accessed separately by a central processing unit (CPU) or an operating system (OS) stored on a control display unit (CDU) or on an FMS in communication with the system  100   c . In other embodiments, at least the processors  110   c  and a portion of the memory  112   c  may be implemented in a simulation computer (e.g., flight simulator) or a training system. 
     In an exemplary embodiment, the single unit  124  may be communicatively coupled with other components of system  100   c  via one or more ports  126 . In some embodiments, the one or more ports  126  may be implemented as a single, bi-directional port. For example, a fiber optic coupler such as a 3 dB, 50:50 (coupling ratio) bidirectional coupler may be used. In other embodiments, the one or more ports  126  may include multiple (e.g., two or more) ports. 
     In an exemplary embodiment, the processor  110  of system  100  may utilize functions or logical instructions and the navigational data received from the memory  112  to generate an output or a reference signal  128 . The output or reference signal  128  may include control parameters such as a roll parameter, a pitch parameter, a bank angle required (BAR) parameter, a delta BAR parameter (ΔBAR), a flight path angle (FPA) parameter, a delta FPA parameter (ΔFPA), or combinations thereof. The output  128  may be received by the control system  116   c  to generate an input or control signal  130  for one or more actuators  132 . For example, the one or more actuators  132  may include a hydraulic system (e.g., hydraulic power pack), a servo motor, an engine, a brake, a control surface such as an aileron, rudder, or elevator, or combinations thereof. 
     Referring now to  FIG. 5  a block, notional flow diagram is shown depicting flow between a guidance module  134 , a navigation module  114  (e.g., including the sensors), a control system module  132 , a first input  136 , an aircraft dynamics module  138 , and a second input  140 . In an exemplary embodiment, the first input  136  comprises at least one of an active flight plan and a planned descent rate derived from the NDB data and updated based on the second input  140 . In this regard, the contents of the reference signal  128  may depend on the first input  136 . For example, if the first input  136  includes the active flight plan, then the reference signal  128  may include the roll parameter, the BAR parameter, the ΔBAR parameter, or combinations thereof. By way of another example, if the first input  136  includes the planned descent rate, then the reference signal  128  may include the pitch parameter, the FPA parameter, the ΔFPA parameter, or combinations thereof. In some embodiments, the first input  136  includes both the active flight plan and the planned descent rate. 
     In an exemplary embodiment, the guidance module  134  may include one or more modules, sub-modules, blocks, sub-blocks, or combinations thereof. In some embodiments, the one or more modules may be a first module  142   a , a second module  142   b , or a combination thereof depending on the first input  136 . For example, referring now to  FIG. 6 , a first module  142   a  of the guidance module  134  may include a proportional-derivative (PD) controller configured to receive the active flight plan from the first input  136  and determine a first one or more deviations  144   a , the first one or more deviations  144   a  being a deviation from the active flight plan. Using the first one or more deviations  144   a , a first controller gain  146 , a second controller gain  148 , and a first set-point  150  (e.g., steady-state value), the first module  142   a  may be configured to determine an output or reference signal  128  (e.g., roll, BAR, ΔBAR, etc.) for the control system  116 . For instance the output or reference signal  128  may be used to re-direct the aircraft  102  according to the active flight plan. By way of another example, a second module  142   b  of the guidance module  134  may include a proportional-integral (PI) controller configured to receive the planned descent rate from the first input  136  and determine a second one or more deviations  144   b  including a deviation rate. For instance, the second one or more deviations  144   b  may be a deviation from a planned descent rate. In this regard, using the second one or more deviations  144   b , an actual descent rate received from the second input  140 , a third controller gain  152 , a fourth controller gain  154 , and a second set-point  156  (e.g., vertical acceleration detection), the second module  142   b  may be configured to determine the output or reference signal  128  (e.g., pitch, FPA, ΔFPA, etc.) for the control system  116 . 
     It is noted that in some embodiments, the first and second module  142   b  of the guidance module  134  may utilize some, all, or none of the same hardware and software. In other embodiments, the first and second module  142   b  of the guidance module  134  may be configured to utilize at least separate software and/or firmware depending on the first input  136 . It is further noted that in some embodiments, the first input  136  may include both the planned descent rate and the aircraft flight plan such that the output or reference signal  128  may include a blend of (i) at least one of a roll, BAR, and ΔBAR, and (ii) at least one of a pitch, FPA, and ΔFPA. 
     In an exemplary embodiment, the guidance module  134  may be tuned according to the specifications of the platform  102  (e.g., aircraft dynamics from module  138 —recognizing that each aircraft is unique). For example, the first controller gain  146 , second controller gain  148 , and first set-point  150  may be specific to a first aircraft, such that the fifth controller gain  158 , the sixth controller gain  160 , and the third set-point  162  may be specific to a second aircraft and its individual aircraft dynamics. 
     In an exemplary embodiment, the guidance module  134  may be tuned according to the guidance mode and/or the specifications of the platform  102 . For example, the first controller gain  146 , second controller gain  148 , and first set-point  150  may be specific to a first guidance mode of an aircraft, such that the seventh controller gain  164 , the eighth controller gain  166 , and the fourth set-point  168  may be specific to a second guidance mode of the aircraft. It is noted that other combinations of controller gains are contemplated by the inventive concepts of this disclosure. For example, other controller gains may include a controller gain that is specific to the first aircraft, the first guidance mode, and the first aircraft dynamics. Other combinations of controller gains encompassed by the inventive concepts disclosed here will be recognized by those skilled in the art. 
     In an exemplary embodiment, the guidance module  134  may be configured to track the defined path  108  using the curved ground path  104  and a reference point. For example, the guidance module  134  may be configured to track the defined path  104  through an RF leg and RF leg transition. In some embodiments, the RF leg may be associated with the LNAV function (e.g.,  FIG. 7 , below). In other embodiments, the RF leg may be associated with the VNAV function ( FIGS. 11-13 , below). In yet other embodiments, the RF leg may be associated with both the LNAV and the VNAV function. 
     Lateral Curves 
     Referring now to  FIG. 7 , the system  100  may be configured to track the defined path  108  through a TF-RF-TF leg sequence (e.g., first Track-to-Fix (TF) leg to second TF leg) using the LNAV function of system  100 . The defined path  108  may be tracked according to the first and second lateral thresholds  106   a ( 1 ) and  106   a ( 2 ), establishing the desired path  104   a , but due to a first one or more deviations  170   a , PDE, and/or PSE, the aircraft  102   a  may be traveling along course  120   a.    
     In an exemplary embodiment, the first and second lateral thresholds  106   a ( 1 ) and  106   a ( 2 ) are each twice their respective RNP value (e.g., 2 NM for RNP-1.0) up to a first point  172  (e.g., segment initial fix) and through to a second point  174  (e.g., segment terminating fix), establishing an outer lateral containment region where there is 99.999% probability that TSE resulting from integrity deviations will be within 2×RNP value. In some embodiments, an inner lateral containment region is established where there is 95% probability that TSE resulting from integrity deviations will be within 1×RNP value. 
     In an exemplary embodiment, multiple containment regions may be defined for the desired path  104  that tracks the defined path  108 . For example, a first containment region may be defined by the first threshold  106   a ( 1 ), the second threshold  106 ( 2 ), and by the portion of the defined path  108  preceding point  172 . By way of another example, a second containment region may be defined by the first threshold  106   a ( 1 ), the second threshold  106 ( 2 ), and by the portion of the defined path following point  174 . By way of yet another example, a third containment region may be defined by the first threshold  106   a ( 1 ), the second threshold  106 ( 2 ), and by the portion of the defined path between points  172  and  174 . 
     In an exemplary embodiment, distances of the first threshold  106   a ( 1 ) and the second threshold  106   a ( 2 ) may vary depending on a respective containment region for which they are associated. For example, within the first and second containment regions discussed above, distances associated with the first threshold  106   a ( 1 ) and the second threshold  106   a ( 2 ) may be equal for both the first and second containment regions, however, within the third containment region, distances associated with the first threshold  106   a ( 1 ) and the second threshold  106   a ( 2 ) may be smaller than the distances associated with the first and second containment regions (e.g., implying a stricter RNP requirement for the third containment region). 
     In an exemplary embodiment, a pre-roll segment  176  may be defined in order to compensate for one or more control deviations. For example, control commands may be issued, but a delay may exist before movement results from the issued control commands. Thus, a pre-roll segment  176  may be defined during a portion of a TF leg such that the delay is compensated and motion occurs precisely when intended during a succeeding RF leg. In some embodiments, the pre-roll segment  176  is a function of multiple parameters including, but not limited to, turn radius (e.g., R1, below), speed, aircraft dynamics, or combinations thereof. In other words, the pre-roll segment  176  may be a function of BAR. In this regard, the BAR may be a function of wind, turn radius, defined path course at a current aircraft location, true airspeed, and ground speed. 
     In an exemplary embodiment, during the pre-roll segment  176 , an integrity deviation may be used to maintain a threshold (e.g.,  106   a ( 1 ) and/or  106   a ( 2 )), while the control deviation may be used by the control system  116  for a control command determination. For example, the PD control module  142   a  may determine a roll command based on a control deviation determined during the pre-roll segment  176 . In some embodiments, outside of the pre-roll segment  176 , an integrity deviation may be equivalent to a control deviation. 
     In an exemplary embodiment, the reference point  178   a  may coincide with an arc having a first radius to a turn center, R1, and distances from the platform  102   a  to the reference point  178   a  may vary depending on a position of the platform  102   a  relative to the desired path  104   a . For instance, if the aircraft  102   a  is at an outer bound of the defined path  104   a , the distance to the reference point  178   a  may be determined as R1+(2×RNP value). If the aircraft  102   a  is at the inner bound, then the distance may be R1−(2×RNP value). If the aircraft  102   a  is at a center of the desired path  104   a  (i.e., in-line with defined path  108 ), then the distance to the reference point  178   a  may be R1. It is noted that the RNP value may depend on the segment of the defined path being traversed. For instance, if the flight segment being traversed is one of the initial, intermediate, and missed approach segments using the LNAV function, then the RNP value may be RNP-1.0; and if the flight segment traversed is the final approach segment, then the RNP value may be RNP-0.3. 
     In an exemplary embodiment, the reference point  178   a  may be determined and used by the processor  110  for performing a level and coordinated turn. For example, referring now to  FIG. 8 , a free-body diagram depicts forces acting on an aircraft during the level and coordinated turn around a turn center point, where a first vector may represent lift, a second vector may represent mass multiplied by centripetal acceleration, a third vector may represent mass multiplied by gravitational forces, with each vector being depicted relative to a portion  180  (e.g., wing) of the platform  102 . 
     In an exemplary embodiment, control commands determined during a level and coordinated turn are determined relative to a curved ground track. For example, velocity vectors and other parameters used in determining BAR are determined relative to the ground. For instance, referring now to  FIG. 9 , radius, R G , and acceleration, a G , used in determining BAR are depicted relative to reference point  178   a  of the curved lateral ground track. In this regard,  FIG. 9  depicts these parameters relative to similar parameters used in conventional systems, which determine these similar parameters relative to an air mass. 
     In an exemplary embodiment, due to the relationship of the velocity vectors, Pythagorean&#39;s theorem, the cosine rule, and/or other trigonometric functions, the control parameters for the level and coordinated turn may be determined while accounting for a measured, detected, or otherwise determined first one or more deviations  170   a  (e.g., lateral drift due to wind). For example, the system  100  may detect a first discontinuity  182   a  (e.g.,  FIG. 7 ) in the flight path, where the first discontinuity  182   a  may be detected by determining an angular difference between a first leg and a second leg. For instance the angular difference could be represented by the curved portion spanning and interposed between points  172  and  174 , and could be determined from input from the NDB. Upon detecting the first discontinuity  182   a , the system  100  may be configured to calculate the BAR for a level and coordinated turn using the law of cosines, the velocity vectors of  FIG. 9 , acceleration, a G , radius R G , and the reference point  178   a  (e.g., turn center), each of which are relative to the ground, according to the following: 
                     cos   ⁡     (     β   trk     )       =           (   TAS   )     2     +       (     G   ⁢   S     )     2     -       (   W   )     2         2   ⁢     (     T   ⁢   A   ⁢   S     )     ⁢   G   ⁢   S               (   1   )                 a   G   =a *cos(β trk )= g   z  tan(φ)cos(β trk )  (2)
 
                     ω   G     =       G   ⁢   S       R   G               (   3   )                 a   G     =           (     G   ⁢   S     )     2       R   G       =           (       R   G     ⁢     ω   G       )     2       R   G       =       R   G     ⁢     ω   G   2                   (   4   )                 ω   G   2     =         a   G       R   G       =         g   z     ⁢     tan   ⁡     (   φ   )       ⁢     cos   ⁡     (     β   trk     )           R   G                 (   5   )                 φ     b   ⁢   i   ⁢   a   ⁢   s       =     φ   =       tan     -   1       ⁡     (         (     G   ⁢   S     )     2         g   z     ⁢     R   G     ⁢     cos   ⁡     (     β   trk     )           )                 (   6   )               
where W is the wind speed in knots, TAS is the true airspeed in knots, GS is ground speed of the aircraft (e.g., platform  102   a ) in knots, β trk  is a tracking drift angle as depicted in  FIG. 9 , R G  is R1 and it is the specified turn radius of a circular ground track in NM, g z  is the gravitational force in NM/hr 2 , coy is an angular turn rate along the circular ground track, and ϕ bias  s the reference roll angle (BAR) in radians for a coordinated turn with level flight to accurately follow the circular ground track (e.g., path  104   a ).
 
     It is noted that Equation (1) above may be substituted into Equation (6) to compute φ bias , Equation (5) above may be found by substituting Equation (2) into Equation (4), and Equation (6) may be found by substituting Equation (3) into Equation (5) to eliminate ω G . 
     In an exemplary embodiment, Equation (1) is valid for specific conditions. For example, Equation (1) may be valid for wind having directions of 0&lt;β trk &lt;180 degrees, because the cosine rule is applicable to triangles. Thus, in some embodiments, a different relationship is defined. For example, the different relationship may be defined where conditions exist with tail wind, with no tail wind, and/or where head wind must be otherwise defined. For instance, when there is no tail wind or there is only tail wind β trk =0 degrees, and when there is head wind β trk =180 degrees. Based on the foregoing conditions, the reference roll angle ϕ bias  (BAR) may be determined according to the following: 
                     φ     b   ⁢   i   ⁢   a   ⁢   s       =       tan     -   1       ⁡     (         (     G   ⁢   S     )     2         g   z     ⁢     R   G         )               (   7   )               
LNAV Function
 
     In an exemplary embodiment, the first module  142   a  of the processor  110  may be configured to determine one or more control commands (e.g., a roll command, etc.) using the LNAV function as follows:
 
γ=− K   p *Deviation L   −K   d *( s *Deviation L )+γ ss   (8)
 
where K p  and K d  are first and second controller gains, γ ss  is a first steady-state value, s denotes a Laplace operator such as a derivative operation.
 
     In an exemplary embodiment, the lateral deviation in Equation (8) is computed based on a direction of a turn or a direction of the RF leg being performed. For example, the right side of the defined path  108  may remain positive according to the following: 
                     Deviation   L     =     {             TurnRadius   -     d   AC       ,     Right   ⁢           ⁢   Turns                     d   AC     -   TurnRadius     ,     Left   ⁢           ⁢   Turns                       (   9   )               
where d AC  is a distance between the actual platform (e.g., aircraft) location and the turn center of a curved ground path.
 
     In some embodiments, the PD controller may be useful to compensate or wash out the first one or more deviations  170   a  according to a deviation rate such that a reference command (e.g., γ from Equation (8)) eventually converges to a required steady-state value (e.g., γ ss  from Equation (8)). It is noted that in Equation (8), the term s*Deviation represents the cross track velocity (CTV), which may be determined according to the following:
 
 CTV=GPSGS *sin( TAE )  (10)
 
where the term GPSGS is the GPS determined ground speed, TAE is defined as Track Angle Error=VTA−Defined Path Course (e.g., path  108 ), where within the TAE definition the term VTA is a GPS defined velocity track angle or an IRU defined velocity track angle.
 
     In some embodiments, a filter is used to remove a bias or a noise from the CTV computation of the LNAV function. For example, a GPS VTA used to compute the TAE may be based on the WGS-84 ellipsoidal earth model, while the Defined Path Course may be based on a Spherical Earth Model. Because the two models are different, a resulting TAE may be offset with a first amount of bias. By way of another example, if the VTA used to compute the TAE is received from an Inertial Reference System (IRS) such as an IRU, the resulting TAE may be offset with a second bias or noise. Thus, a filter as shown in  FIG. 10A  may be used to filter the CTV to remove the unwanted bias. 
     In an exemplary embodiment, the filter used to filter CTV may need to manage multiple filters, filter one or more signals from different sources, or be configured to compensate for multiple types of bias and/or noise. For example, high-frequency noise/bias resulting from a derivative performed to determine the first one or more deviations  170   a  (e.g., lateral deviation) may result high-frequency noise that is filtered/compensated by a first low-pass filter. By way of another example, an accelerometer of an IRU may result low-frequency noise that is filtered by a second high-pass filter. Thus, the filter used to filter the CTV may be the notional complementary filter of  FIG. 10A , useful in managing multiple (e.g., two or more types of filters (e.g., low-pass filters and high-pass filters). 
     In an exemplary embodiment, an output of the notional complementary filter of  FIG. 10A  may be obtained according to the following:
 
 X ( s )= S ( s )+ N   1 ( s ) G ( s )+ N   2 ( s )[1− G ( s )]  (11)
 
where G(s) and 1−G(s) represent transfer functions, which do not impact the actual signal. However, these transfer functions are useful in showing how one or more inputs and outputs are related. For example, the two noise inputs, N 1  and N 2 , are modified by the transfer functions G(s) and 1−G(s), respectively, meaning that the two noises, N 1  and N 2 , may have complementary spectral characteristics, such that G(s) can be chosen to mitigate the noise in both inputs. For instance, N 1  may be predominantly high-frequency noise from the lateral deviation derivative, whereas N 2  may be predominantly low-frequency noise due to the accelerometer.
 
     Referring now to  FIG. 10B , the transfer functions G(s) may be chosen according to the following: 
                         X     c   ⁢   t   ⁢   v       ⁡     (   s   )         c   ⁢   t   ⁢     v   ⁡     (   s   )           =         τ   ctv     s         τ     c   ⁢   t   ⁢   v         s   +   1                 (   12   )                     X     L   ⁢   D       ⁡     (   s   )         L   ⁢     D   ⁡     (   s   )           =     s       τ     c   ⁢   t   ⁢   v         s   +   1                 (   13   )               
where τ ctv  is a time constant based on the cut-off frequency for the filter. For example, the time constant may be 15 seconds for a cut-off frequency of 0.0667 (i.e., 1/15) radians/sec for a low pass filter. It is noted that other time constants for respective cut-off frequencies are also contemplated by the inventive concepts disclosed herein.
 
     In an exemplary embodiment, two redundant measurements of the same signal may be used by the complementary filter. For example, the first redundant measurement may include CTV from Equation (10), while a second redundant measurement may include a derivative operation used to determine the lateral deviation. In some embodiments, the redundant measurements are input into a complementary filter (e.g.,  FIG. 10B ) to blend the signals and generate an estimate of the actual signal. In other embodiments, the redundant measurements may be input into a Kalman filter, a Wiener filter, or other filters known in the art. 
     In an exemplary embodiment, a discrete time output of the complementary filter of  FIG. 10B  may be obtained according to the following:
 
 x ( t )= x   ctv ( t )+ x   LD ( t )  (14)
 
where x ctv (t) is a first estimated CTV due to the input of ctv(t), x LD (t) is a second estimated CTV due to the input of LD(t), x(t) is the filtered CTV.
 
     In an exemplary embodiment, the one or more transfer functions, G(s) (e.g., Equations (12) and (13)), are used to convert the output of the filter for filtering CTV from the Laplace domain to the discrete time domain. 
     In an exemplary embodiment, the filtered CTV may then be input into Equation (8). In order to input them into the first module  142   a  (e.g., input as embedded code into the PD controller) they must first be converted into the discrete time domain. For example, the transfer functions of Equations (12) and (13) are used to convert from the Laplace Domain to the discrete time domain. For instance, equation (13) may be rearranged according to the following:
 
(τ ctv   s+ 1) X   LD ( s )= sLD ( s )  (15)
 
where Equation (13) may be further manipulated by performing the inverse Laplace transform with all initial conditions set to zero according to the following:
 
                         τ   ctv     ⁢       sX     L   ⁢   D       ⁡     (   s   )         =       s   ⁢   L   ⁢     D   ⁡     (   s   )         -       X     L   ⁢   D       ⁡     (   s   )           ⁢     
     ⁢         x   .     ⁢     LD   ⁡     (   T   )         =         1     τ     c   ⁢   t   ⁢   v         ⁢       d   ⁡     (     L   ⁢     D   ⁡     (   t   )         )       dt       -         x     L   ⁢   D       ⁡     (   t   )         τ     c   ⁢   t   ⁢   v                     (   16   )               
where a numerical integration method (e.g., Euler&#39;s method, trapezoidal method, etc.) may be used to re-write Equation (16) in the discrete time domain according to the following:
 
                             x   LD     ⁡     (   k   )       -       x   LD     ⁡     (     k   -   1     )           Δ   ⁢           ⁢   T       =           LD   ⁡     (   k   )       -     LD   ⁡     (     k   -   1     )           τ     ctv     Δ   ⁢           ⁢   T           -         x   LD     ⁡     (     k   -   1     )         τ   ctv           ⁢     
     ⁢         x   LD     ⁡     (   k   )       =         1     τ   ctv       ⁢     (       LD   ⁡     (   k   )       -   LD   -     (     k   -   1     )       )       +       (     1   +       Δ   ⁢           ⁢   T       τ   ctv         )     ⁢       x   LD     ⁡     (     k   -   1     )                     (   17   )               
where ΔT is proportional to the rate at which the LNAV function is run, and k is a time step index that is specific to a particular flight leg sequence. For example, with respect to ΔT, if the LNAV function is run at a rate of 10 Hz, then ΔT may be 0.1 seconds, and if the LNAV function is run at 60 MHz, then ΔT may be 6*10 −7  seconds.
 
     In an exemplary embodiment, Equation (12) is manipulated similarly to Equation (13). For example, Equation (12) may be manipulated according to the following: 
                         (         τ   ctv     ⁢   s     +   1     )     ⁢       X   ctv     ⁡     (   s   )         =       τ   ctv     ⁢     s   ⁡     (     ctv   ⁡     (   s   )       )           ⁢     
     ⁢         x   .     ctv     =         d   ⁡     (     ctv   ⁡     (   t   )       )       dt     -         x   ctv     ⁡     (   t   )         τ   ctv           ⁢     
     ⁢         x   LD     ⁡     (   k   )       =       ctv   ⁡     (   k   )       -     ctv   ⁡     (     k   -   1     )       +       (     1   -       Δ   ⁢           ⁢   T       τ   ctv         )     ⁢       x   ctv     ⁡     (     k   -   1     )                     (   18   )               
where Equations (14), (17), and (18) may be used to determine the filtered CTV in the discrete time domain according to the following:
 
     
       
         
           
             
               
                 
                   
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                         ⁢ 
                         
                           v 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     It is noted that Equation (19) may be used to compute the filtered CTV in the discrete time domain, where x(k) is the filtered CTV at a time step k within a particular leg sequence. It is further noted that in the derivation of Equation (19) x(t) is assumed to be in units of meters per second (m/s), ctv(t) is in m/s, and LD(t) is in meters. The use of these units is not meant to be limiting, and other units may be used together with proper unit conversions. 
     Leg Transitions 
     In an exemplary embodiment, the first module  142   a  and/or the second module  142   b  of processor  110  may reset a value of the CTV. For example, the TAE used to compute ctv(k) may experience a larger error than normal due to the platform  102  transitioning from a first leg to a second leg. In order to compensate for the large amount of error, at the beginning of a newly sequenced flight leg, the filtered CTV, x(k), may be reset to capture the jump in TAE. For instance, at the beginning of a second flight leg sequence at an initial time step k, the filtered CTV, x(k), may be computed according to the following, instead of using Equation (19):
 
 x ( k )= ctv ( k )  (20)
 
Vertical Curves
 
     Referring now to  FIG. 11 , the system  100  may be configured to track movement of the platform  102   b  along the defined path  108   b  and through a RF leg transition using the VNAV function of system  100 . The defined path  108   b  may be tracked according to the first and second vertical thresholds  106   b ( 1 ) and  106   b ( 2 ), establishing the desired path  104   b , but due to a second one or more deviations  170   b  (e.g., from disturbances due to wind, PDE, and/or PSE), the aircraft  102  may be traveling along course  120   b . In an exemplary embodiment, the first and second vertical thresholds  106   b ( 1 ) and  106   b ( 2 ) are each approximately 500 ft. (153 m), establishing a first vertical containment region where there is 99.7% probability that TSE resulting from vertical deviations will be within the RNP value. It is noted that the first vertical containment region may be applicable for any path segment that is not a Final Approach Segment (FAS). During the FAS, the first and second vertical thresholds  106   b ( 1 ) and  106   b ( 2 ) may be 75 ft. (23 m), establishing a second vertical containment region. 
     In an exemplary embodiment, the reference point  178   b  may coincide with an arc having a second radius, R2, and distances from the platform  102   b  to the reference point  178   b  may vary depending on a position of the platform  102   b  relative to the desired path  104   b.    
     In an exemplary embodiment, the system  100  may determine a second discontinuity  182   b  (e.g., vertical transition between level flight to descent) in the defined path  108   b . Upon determining the second discontinuity  182   b , the system  100  may determine the top of descent (TOD) point  184  ( FIG. 12 ) and/or the reference point  178   b  for performing the transition from the level path (e.g., cruise or Altitude Hold Mode) to the descent (e.g., Descent Mode). For example, referring now to  FIGS. 11 and 12 , a vertical discontinuity  182   b  may exist between a first leg  186  and a second leg  188 . Rather than determining control parameters relative to the first leg  186  or the second leg  188  to transition from the first leg  186  to the second leg  188 , control parameters may be determined relative to an arc  190  that connects the first leg  186  with the second leg  188  and relative to a reference point  178   b  of the arc  190 . Further, multiple modes of the multi-mode controller (e.g., control system  116 ), may be utilized to smoothly transition between the first leg  186  and the second leg  188 . 
     In an exemplary embodiment, the multiple guidance modes of the multi-mode controller  116  include, but are not limited to, a Lateral Path (LPATH) Mode, a Vertical Path (VPATH) Mode, a Descent Mode, an Altitude Hold Mode  192 , a Capture Mode  194 , a Transition to Track Mode  196 , and a Track Mode  198 . One or more of the multiple modes may be utilized by the multi-mode controller to conduct navigational guidance for different segments of the defined path  108 . For example, the different segments may include different flight phases such as Cruise, Descent, and Approach, where these phases may have separate sub-phases (e.g., Final Approach). 
     In an exemplary embodiment, the radius R2 of the arc  190  may be determined according to the following: 
                     R   ⁢   2     =       T   ⁢   A   ⁢     S   2         G   ⁢   1               (   21   )               
where TAS is the true airspeed in knots and G1 is a specified g-force experienced during a Capture Mode  194  maneuver. In some embodiments, G1 is a first user-defined parameter ranging from 0.05 g to 0.1 g. In yet other embodiments, an actual value of the g-forces experienced may be higher than a defined parameter (e.g., G1) due to wind forces.
 
VNAV Function
 
     In an exemplary embodiment, the second module  142   b  of the processor  110  may be configured to determine one or more control commands (e.g., the delta FPA command, a pitch command, etc.) using the VNAV function as follows: 
                     Δ   ⁢   γ     =           -     K   p       *   D   ⁢   e   ⁢   v   ⁢   i   ⁢   a   ⁢   t   ⁢   i   ⁢   o   ⁢     n   V       -       K   i     *     (       D   ⁢   e   ⁢   v   ⁢   i   ⁢   a   ⁢   t   ⁢   i   ⁢   o   ⁢     n   V       s     )       -     γ   VAD         max   ⁡     (       V   g     ,       min_   ⁢   V     g       )                 (   22   )               
where K p  and K i  are controller gains, γ VAD  is a vertical acceleration detection, and 1/s denotes a Laplace operator such as an integral operation, and Deviation V  is the vertical deviation rate. It is noted that the term γ VAD  of Equation (22) is only used in the Track Mode  198  of the control system  116 , and is useful for detecting and compensating for sudden vertical movement induced by deployment of aircraft control surfaces. This sudden deployment can increase the second one or more deviations  170   b , and is therefore accounted and compensated for in Equation (22).
 
     In an exemplary embodiment, the PI controller may use linearized input and may be useful to eliminate an offset from the second one or more deviations such that a command from reference signal  128  (e.g., Δγ from Equation (22)) eventually converges to a required steady-state value (e.g., γ VAD  from Equation (22)). It is noted that in Equation (22), the term 
               D   ⁢   e   ⁢   v   ⁢   i   ⁢   a   ⁢   t   ⁢   i   ⁢   o   ⁢     n   V       s         
represents the vertical deviation (VD) (e.g.,  170   b ), which is in nautical miles (NM). It is further noted that in conventional systems, a filter is used to compensate for a large discontinuity once the descent path has been captured. For example, a low-pass filter (LPF) may be used for VNAV in conventional systems according to the following:
 
                     Δ   ⁢   γ     =       L   ⁢   P   ⁢     F   ⁡     (         -     K   i       *   V   ⁢   D     -       K   p     *   V   ⁢   D   ⁢   R       )           max   (       V   g     ,     min_V     g   )                     (   23   )                 VD=alt   current   −alt   planned   (24)
 
 VDR=V   s   −V   s_planned   (25)
 
where K p  and K i  are controller gains (e.g., K i =180 hr −1  and K p =1.0), LPF is the low-pass filter operation (e.g., τ of 20 seconds), VD is vertical deviation in NM, V g  is ground speed in knots, alt current  is the current altitude of the platform  102   b  (e.g., aircraft) in NM, alt planned  is the planned altitude in NM, V s  is the current vertical speed of the platform  102   b  in knots, and V s_planned  is the planned vertical speed in knots.
 
     In conventional systems, the VNAV function of Equation (23) uses the LPF operation to account for a large vertical deviation as the function transitions between the Altitude Hold Mode  192  to the Capture Mode  194 . In exemplary embodiments, using the enhanced guidance laws disclosed herein, the large vertical deviation is avoided, negating the need for the LPF operation of the conventional systems (e.g., as shown in Equation (22) above). For example, the large vertical deviation may be avoided by calculating the second one or more deviations  170   b  relative to the arc  190 , as discussed with respect to Equation (21) above. 
     Altitude Hold Mode 
     In an exemplary embodiment, the system  100  performs level flight in the Altitude Hold Mode  192 . During the Altitude Hold Mode  192 , the ΔFPA may be determined according to the following: 
                     Δ   ⁢   γ     =       (         -     K   i       *   V   ⁢   D     -       K   p     *   V   ⁢   D   ⁢   R       )       max   ⁡     (       V   g     ,     min_V   g       )                 (   26   )               
where VDR is the vertical deviation rate in knots, VD is a vertical deviation (e.g., second one or more deviations  170   b ) in NM, K i  ranges from 810 hr −1  to 180 hr −1 , K p  is 0.8, V g  is the ground speed in knots, and min_Vg is the specified minimum ground speed in knots.
 
Capture Mode
 
     In an exemplary embodiment, the system  100  transitions from the Altitude Hold Mode  192  to the Capture Mode  194 . For example, the transition may occur upon detecting a FPA change, such as the deviation  170   b , above an Altitude Mode threshold, where the threshold may be ΔFPA to be greater than a specific value. For instance, for a two degree)(2° ΔFPA, a transition may occur. In contrast, a conventional system may enter Capture Mode upon detecting a 250-500 ft. (76-153 m) deviation from a descent leg, which results in a large vertical deviation on a flight display before the transition. 
     Referring now to  FIGS. 12-13 , the system  100  begins the transition to Capture Mode  194  relative to a beginning point  204  of the arc  190 , interposed between, spanning, or connecting two different legs (e.g., third TF leg and fourth TF leg) of the defined path  108 . 
     In an exemplary embodiment, in order to determine point  204 , the second discontinuity  182   b  or angle between flight legs in a flight path (e.g., defined path  108   b ) may be detected and determined according to the following:
 
β=π−|γ 1 −γ 2 |  (27)
 
where γ 1  is the FPA of the first leg  186  and γ 2  is the FPA of the second leg  188  (e.g., where first leg  186  and second leg  188  may be juxtaposed portions of the defined path  108 ). If the second discontinuity  182   b  is vertical and is above the Altitude Hold threshold, then the system  100  may enter Capture Mode  194 . In some embodiments, the discontinuity  182   b  is converted from radians to degrees.
 
     Referring now to  FIG. 13 , using the radius, R2, and the second discontinuity angle, B (from Equation (27)), a distance from the TOD  206  to beginning point, or first tangent point, T1  204  of the arc  190  may be determined according to the following: 
     
       
         
           
             
               
                 
                   
                     distance 
                     ⁡ 
                     
                       ( 
                       
                         
                           T 
                           1 
                         
                         , 
                         TOD 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       R 
                       ⁢ 
                       2 
                     
                     
                       tan 
                       ⁡ 
                       
                         ( 
                         
                           β 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In an exemplary embodiment, a planned descent speed, V s_planned , corresponding to the current aircraft  102   b  location on the vertical arc  190  may be determined according to the following:
 
 V   s_planned   =V   g *tan(θ)  (29)
 
where theta, θ, may be determined according to the following:
 
If |γ1|&lt;|γ2| (see, e.g.,  FIG. 13 ),
         Step 1: calculate a distance ζ between point  208  and point T1  204  according to the following:
 
ζ= R 2*sin( aγ   1 )  (30)
   where aγ 1 =|γ 1 |.   Step 2: calculate horizontal distance, e, between point  206  and point  204  according to the following:
 
 e=d *cos( aγ   1 )  (31)
   where d in Equation (31) is the distance calculated from Equation (28).   Step 3: assign proper sign according to the following:
 
Capture Sign =1
 
Else if |γ1|&gt;|γ2|
   Step 1: calculate a distance ζ between point  208  and point T2  210  (e.g., second arc tangent point coinciding with second leg) according to the following:
 
ζ= R 2*sin( aγ   2 )  (32)
   where aγ 2 =|γ 2 |.   Step 2: calculate distance, e, between point  210  and point  206  according to the following:
 
 e=d *cos( aγ   1 )  (33)
   where d in Equation (33) is the distance calculated from Equation (28).   Step 3: assign proper sign according to the following:
 
Capture Sign =−1
 
end if
   Step 4: determine horizontal distance from point  208  to last waypoint on flight path (e.g., point  202  of  FIG. 12 ) according to the following:
 
 d   208   end   =d   206   end +Capture Sign *( e +ζ)  (34)
   where d 206   end  is the horizontal distance from point  206  to the last waypoint on the flight path (e.g., point  202 ).   Step 5: determine the distance between point  212  and the aircraft  102   b  according to the following:
 
 f =Capture Sign *( d   208   end   −d   aircraft   end )  (35)
   where d aircraft   end  is the horizontal distance from the aircraft  102   b  to the last waypoint on the flight path (e.g., point  202 ).   Step 6: calculate the vertical distance from reference point  178   b  to point  212  according to the following:
 
distance(212,178 b )=√{square root over ( R   2   −f   2 )}  (36)
   Step 7: determine the planned vertical speed, V s_planned , using the distance from Equation (36) is the following:       

     
       
         
           
             
               
                 
                   
                     V 
                     
                       s 
                       planned 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           V 
                           g 
                         
                       
                       * 
                       f 
                     
                     
                       distance 
                       ⁡ 
                       
                         ( 
                         
                           212 
                           , 
                           
                             1 
                             ⁢ 
                             7 
                             ⁢ 
                             
                               8 
                               b 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     In some embodiments, a planned altitude, alt _planned , along the arc  190  may be determined. For example, the planned altitude may be determined as additional steps after Step 7 above, according to the following:
         Step 8: determine altitude at the reference point  178   b  of the arc  190  according to the following:       

                       alt     1   ⁢   7   ⁢   8   ⁢   b       =       alt     2   ⁢   0   ⁢   6       +       Capture     S   ⁢   i   ⁢   g   ⁢   n       *     (       e   *     tan   ⁡     (   γ   )         -     R   *     cos   ⁡     (   γ   )           )           ⁢     
     ⁢     γ   =     {             a   ⁢           ⁢     γ   1       ,           ⁢         wh   ⁢   en     ⁢           ⁢          γ   1            &lt;          γ   2                          a   ⁢     γ   2       ,           ⁢         wh   ⁢   en     ⁢           ⁢          γ   1            &gt;          γ   2                                (   37   )               
where, alt 206  is the altitude at point  206  or the TOD.
         Step 9: solve for the planned altitude, alt _planned , according to the following:
 
 alt   planned   =alt   178b +Capture Sign *distance(212,178 b )  (38)
 
Transition to Track Mode
       

     In an exemplary embodiment, the system  100  switches from the Capture Mode  194  to the Transition to Track Mode  196 . Operating in the Transition to Track Mode  196  may include using a low-pass filter to fade or blend bias as the system  100  transitions from the Capture Mode  194  to the Track Mode  196 . For example, a bias associated with a FPA command determined at the end of the Capture Mode  194 , may be faded or blended with a FPA command computed by system  100  at the beginning of the Track Mode  196 . For instance, the low-pass filter in the Transition to Track Mode  196  may be initialized with bias according to the following:
 
Δγ cap2trk (0)=Δγ trk −Δγ cap   (39)
 
where Δγ trk  is the ΔFPA command computed with the VNAV function at the beginning of the Track Mode  196 , and Δγ cap  is the ΔFPA command computed with the VNAV function at the end of the Capture Mode  194 .
 
     In an exemplary embodiment, the ΔFPA command is determined in the Transition to Track Mode  196  after initialization according to the following:
 
Δγ cap2trk ( t )= LPF (Δγ cap2trk ( t− 1))
 
Δγ( t )=Δγ trk ( t )−Δγ cap2trk ( t )  (40)
 
where the LPF has a time constant, τ, ranging from three to eight seconds. For example, the time constant, τ, may be seven seconds.
 
Track Mode
 
     In an exemplary embodiment, the system  100  switches from the Transition to Track Mode  196  to the Track Mode  198 . Operating in the Track Mode  198  may include accurately tracking or following the descent path (e.g., leg  188 ). During the Track Mode  198 , the ΔFPA may be determined according to the following: 
                       Δ   ⁢   γ     =       (         -     K   i       *   V   ⁢   D     -       K   p     *   VDR     -   VAD     )       max   ⁡     (       V   g     ,     min_V   g       )           ⁢     
     ⁢     VAD   =     L   ⁢   P   ⁢     F   ⁡     (         1   .   5     *   V   ⁢   A     +     V   ⁢   D   ⁢   R       )                   (   41   )               
where VA is the vertical acceleration in ft./s 2 , which may be converted to knots (e.g., divide by 6076.11), VDR is the vertical deviation rate in knots (e.g., VDR=actual vertical speed−defined vertical speed), VD is a vertical deviation (e.g., second one or more deviations  168   b ) in NM, K i  ranges from 810 hr −1  to 180 hr −1 , K p  is 0.8, V g  is the ground speed in knots, and min_Vg is the specified minimum ground speed in knots. It is noted that the VNAV function uses the VAD term to compensate for sudden vertical movement induced by deployment of control surfaces via the low-pass filter (LPF), where the LPF has a time constant ranging from four to seven seconds. For example, the time constant may be four seconds.
 
     In an exemplary embodiment, a Track Mode threshold may be set such that a detected FPA discontinuity during the Track Mode  198  that is greater than the Track Mode threshold results in a determination of a control command relative to the arc  190  and a determination of planned descent rate and altitude according to the Capture Mode  194  of system  100 . In an exemplary embodiment, the Track Mode threshold ranges from 0.5 degrees FPA discontinuity to 1.5 degree discontinuity. For example, the Track Mode threshold may be 0.8 degrees FPA discontinuity for determining vertical deviation relative to a vertical arc (e.g., arc  190 ) as opposed to determining vertical deviation relative to a flight leg (e.g., conventional systems determine the deviations relative to the flight legs throughout the entire Track Mode  198 ). 
     Referring now to  FIG. 14  an exemplary embodiment of a method  300  according to the inventive concepts disclosed herein may include one or more of the following steps. For example, the method  300  may be a method for determining a navigational solution using enhanced guidance laws. 
     A step  302  may include receiving, via a processor, at least one input comprising navigation data indicative of a defined path from a navigation database (NDB) and position data from at least one sensor. In some embodiments, the processor is one of multiple processors of a navigational guidance computer, the position data includes a current platform position, and the navigational data includes a planned platform position with respect to a portion of the defined path being traversed. 
     A step  304  may include comparing juxtaposed portions of the defined path to determine a discontinuity in the defined path. For example, the discontinuity may be calculated by comparing a first angle of a first flight leg to a second angle of a second flight leg to determine a change in flight path angle. In some embodiments, the discontinuity comprises a lateral discontinuity. In other embodiments, the discontinuity comprises a vertical discontinuity. 
     A step  306  may include defining, via the processor, a containment region and a curved reference line of a non-linear desired path that tracks the defined path, the containment region defined according to a threshold distance on either side of the defined path, and the curved reference line interposed between the juxtaposed portions to span the discontinuity. For example, the curved reference line may be an arc associated with the non-linear desired path, such as a portion of a circular ground path, and the containment region may be defined according to a first distance between the arc and the defined path and a second distance between the arc and the defined path, where the second distance is greater (e.g., second distance=first distance×two) than the first distance. In this regard, the defined path may be equivalent to a center line of the containment region. 
     A step  308  may include comparing a current platform position derived from the position data to a planned platform position derived from the navigation data to determine a platform deviation relative to the defined path, the desired path, and the curved reference line. 
     A step  310  may include outputting a reference signal for controlling the platform relative to the defined path, the desired path, and the curved reference line, to compensate for the deviation and maintain the platform deviation within the threshold distance. For example, the reference signal may be output to a vehicle control system, such as an autopilot or digital engine control, to control or maneuver the vehicle through a curvature of the curved reference line relative to the curved reference line and a reference point or turn center. The vehicle may be controlled or maneuvered while ensuring that platform deviations are maintained within the threshold distance according to one or more degrees of accuracy. In this regard, the one or more degrees of accuracy may be an RNP value from RNP-0.1 to RNP-1.0. 
     In some embodiments, method  300  may include one or more sub-steps. For example, referring now to  FIG. 15 , method  300  may include one or more or sub-steps, including but not limited to,  306 - 1 ,  308 - 1 ,  310 - 1 ,  310 - 2 , and  310 - 3 . 
     A sub-step  306 - 1  may include determining a radius of a curvature of the curved reference line by selecting a desired g-force associated with maneuvering the platform through the curvature. The selecting may be performed by a user or a manufacturer. For example, the selecting may be via firmware or software embedded in an SOC. By way of another example, the selecting may be via a user interface during real-time operation of the platform. 
     It is noted that sub-step  306 - 1 , or determining of the radius, may be an important sub-step in method  300 . For example, the turn center or reference point for an arc spanning or connecting the discontinuity or FPA of the segments of a flight path may be dependent on the radius selected. In other words, one or more parameters utilized in performing the LNAV Function or the VNAV Function may be dependent on the radius selected. By way of another example, another step dependent on the radius selection may be the Step  308 , which may determine the deviation of the platform from the defined path relative to the arc after the radius of the arc is determined. 
     A sub-step  308 - 1  may include determining a deviation rate based on the position data received in Step  302 . In some embodiments, the deviation rate is determined relative to a non-linear portion of the desired path (e.g., curved reference line or arc). Thus, the deviation rate may be further based on the radius selected in sub-step  306 - 1 , a planned vertical speed of the platform at its current position, and a planned altitude of the platform. In this regard, the planned vertical speed and the planned altitude may be received and/or derived from an NDB or from a flight plan. 
     The sub-step  308 - 1  may include determining the deviation rate together with the deviation determined in step  308 . For example, during the Capture Mode of the FMS, the deviation and the deviation rate may be based on the radius of the curved vertical ground track. Next, the deviation, the deviation rate, and a planned vertical speed at a current position of the aircraft, may be used to determine a delta flight path angle Δγ command according to Equations (27) through (38). 
     By way of another example, the sub-step  308 - 1  may include determining two or more deviations. For example, the first deviation may be a vertical deviation from a curved vertical ground track, the discontinuity may be a ΔFPA, and legs of the ΔFPA may be connected by an arc proportional to the curved ground track. In this regard, the second deviation may be a lateral deviation from a curved lateral ground track. By way of another example, the step  308  may include determining a control deviation, such that sub-step  308 - 1  may include determining one or more integrity deviations. In this regard, the step  310  may include using the control deviation as an input into the LNAV function and using the integrity deviation to determine whether the platform deviations are maintained within the threshold distance during controlling or maneuvering the platform through a non-linear portion of the defined path. For instance, the non-linear portion may be a curved lateral ground track with a first ground-based reference point, such that the pre-roll time (e.g. used to account for control deviation) of a pre-roll segment may be determined according to the following: 
     
       
         
           
             
               
                 
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         BAR 
                       
                       &gt; 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         BAR 
                       
                       ≤ 
                       22 
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       Γ 
                       = 
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             BAR 
                           
                           11 
                         
                         + 
                         4.5 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       TD 
                       TAS 
                       max 
                     
                     = 
                     200 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       TD 
                       TAS 
                       min 
                     
                     = 
                     150 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         TAS 
                       
                     
                     ≥ 
                     
                       TD 
                       TAS 
                       max 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       else 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         TAS 
                       
                     
                     &lt; 
                     
                       TD 
                       TAS 
                       min 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       6 
                       - 
                       Γ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   else 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     c 
                     = 
                     
                       
                         
                           ( 
                           
                             6 
                             - 
                             Γ 
                           
                           ) 
                         
                         * 
                         
                           ( 
                           
                             
                               TD 
                               TAS 
                               max 
                             
                             - 
                             
                               V 
                               TAS 
                             
                           
                           ) 
                         
                       
                       
                         
                           TD 
                           TAS 
                           max 
                         
                         - 
                         
                           TD 
                           TAS 
                           min 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     end 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     if 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Ϛ 
                       en 
                     
                     = 
                     
                       Γ 
                       + 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     else 
     
       
         
           
             
               
                 
                   
                     Ϛ 
                     
                       e 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       P 
                       ⁢ 
                       1 
                     
                     + 
                     
                       
                         
                           0 
                           . 
                           7 
                         
                         ⁢ 
                         5 
                         * 
                         Δ 
                         ⁢ 
                         B 
                         ⁢ 
                         A 
                         ⁢ 
                         R 
                       
                       
                         P 
                         ⁢ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     end if
 
If ζ en &gt;8.5, then ζ en =8.5  (44)
 
where ΔBAR is a difference between a bank angle required (BAR) of a first leg (e.g., next leg) and a BAR of a second leg (e.g., current leg) in units of degrees, Γ and c are temporary variables or symbols and Γ is in units of seconds, TD TAS   max  and TD TAS   min  comprise a velocity threshold in units of knots, ζ en  is the time associated with the pre-roll segment in units of seconds as determined by an enhanced guidance law, and P 1  and P 2  are platform dependent parameters with P 1  being in units of seconds and P 2  being in units of degrees per second. For instance, P 1  may be proportional to a rate of the processor, such as if the processor is a 10 Hz processor, then P 1  may be 1 second, and if the processor is a 60 MHz processor, then P 1  may be 6*10 −6 . In this regard, P 2  may be the nominal roll change rate. For example, P 2  may be 3 degrees per second.
 
     It is noted that the pre-roll time determined above is determined with respect to maintaining the integrity deviation determined in sub-step  308 - 1 . 
     A sub-step  310 - 1  may include determining the reference signal to include an output that accounts for a special condition, such as wind. Thus, in some embodiments, the reference signal determined in Step  310  may only be output after accounting for the special conditions or circumstances, such as wind. For example, the LNAV function may determine one or more control commands (e.g., a roll command) according to Equations (1) through (19), while accounting for wind (e.g., sub-step  310 - 1 ). For example, the BAR or the roll command turn bias, ϕ, may be determined using Equations (1) through (7) according to the following: 
     (a) if wind, then
 
Δψ=ψ p −ψ wind  
 
     (b) if Δψ&gt;180 degrees
 
Δψ=360−Δψ
 
     (c) else if Δψ&lt;−180 degrees
 
Δψ=360+Δψ
 
     (d) if 1&lt;abs(Δψ)&lt;179 degrees, then 
     
       
         
           
             
               
                 
                   Φ 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           2 
                           * 
                           T 
                           ⁢ 
                           A 
                           ⁢ 
                           S 
                           * 
                           
                             
                               ( 
                               
                                 G 
                                 ⁢ 
                                 S 
                               
                               ) 
                             
                             3 
                           
                         
                         
                           
                             g 
                             z 
                           
                           ⁢ 
                           
                             R 
                             G 
                           
                           * 
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     T 
                                     ⁢ 
                                     A 
                                     ⁢ 
                                     S 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     G 
                                     ⁢ 
                                     S 
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   W 
                                   ) 
                                 
                                 2 
                               
                             
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     (e) else 
                   Φ   =       tan     -   1       ⁡     (         (     G   ⁢   S     )     2         g   z     ⁢     R   G         )               (   46   )               
where Δψ is a track angle difference between a track angle of the platform and a wind vector with direction in units of degrees, the BAR or roll command turn bias, ϕ, is in radians, GS is a ground speed in knots, R G  is a turn radius of the defined non-linear path relative to ground, g z  is gravitational force in NM/hr 2 , W is wind speed in knots, TAS is true airspeed in knots, and where (a) above may occur during sub-step  310 - 1 , including accounting for special conditions or circumstances such as wind and (b) may occur during sub-step  310 - 2 , including computing a track angle difference.
 
     It is noted that an assumption is made in determining the BAR or roll command turn bias, ϕ, in Equations (45) and (46) above. The assumption being made is that the track angle of the aircraft is within a certain set, (e.g., ψ ac ∈[−180, 180)) and that the direction of the wind vector is within a certain set (e.g., ψ wind ∈[−180, 180)) with true north being zero (0) degrees. It is further noted that the functions, Δψ=360−Δψ and Δψ=360−Δψ of (a) and (b) above, represent functions to scale the track angle difference to a 180 degree (180°) window. 
     By way of another example, the sub-steps  310 - 1  may include accounting for special conditions or circumstances such as compensating, during the Track mode, for one or more deployed control surfaces. For instance, a deployed control surface may be accounted for and a ΔFPA command determined according Equation (41), above. 
     By way of yet another example, the sub-step  310 - 1  may include accounting for special conditions by negating, during the Altitude Hold Mode, a need for vertical acceleration detection (VAD), and determining the ΔFPA command according to Equation (26). 
     A sub-step  310 - 2  may include filtering a signal associated with at least one of a cross-track velocity, a ΔFPA, a vertical acceleration detection (VAD), and a deviation rate. For example, a bias may be carried from a first ΔFPA determined in a first mode of the FMS to a second mode of the FMS. In the second mode of the FMS, the bias may be filtered to determine a second ΔFPA for the second mode. For instance, the bias may be filtered during the Transition to Track Mode according to Equations (39) through (40). 
     A sub-step  310 - 3  may include applying a flag or data bit to enable a determination that the curvature of the curved reference line indicates a right, a left turn, a descent, or an ascent. In this regard, it is noted that a portion of a defined path, such as a turn segment of a flight path, may include two identical but opposite segments. For example, the turn segments of an S-turn may be equal but opposite to each other with respect to one or more directions at which the turn segments are performed. Thus, in some embodiments, calculations or computations are made for a first segment using a first flag or data bit to indicate a first one or more directions. Then, rather than re-compute or re-calculate functions for the second turn segment, the navigational guidance computer may be configured to re-use one or more previous calculations, while applying a second flag or a second data bit to indicate that the one or more previous calculations will apply to a second one or more directions. 
     In some embodiments, the flag or data bit may indicate a proper sign convention associated with the turn. For instance, when using the LNAV function, one or more equations such as Equation (9) may be applied. By way of another example, sub-step  310 - 3  may include using the VNAV function and applying Step 3 of one or more of Equations (29)-(33). 
     Examples 
     By way of another The following examples are to be considered as illustrative in nature, and are not limiting in any way. The scope of the invention is that which is defined in the claims only. 
     Flight technical error (FTE) is primarily influenced by the guidance laws in an FMS, auto pilot, and auto-throttle. Therefore, to evaluate the statistical performance of the FTE, a simulation tool was developed to simulate and integrate one or more avionics models that can impact the FTE. 
     Referring now to  FIG. 16 , guidance laws for an FMS were implemented as embedded Matlab functions. Matlab/Simulink models for auto pilot and auto-throttle  116   d  were based on production software for Bombardier Global 5000 business jets (a.k.a. M145). In this regard, a six degrees of freedom (DOF) (e.g., three for vertical and three for lateral) model  214  was provided as Simulink S-functions according to the Bombardier Global 5000 business jets. 
     Monte Carlo simulation was performed for each of the following Examples. In this regard, 100 runs were performed for each Example, where the Monte Carlo test driver called the simulation tool in  FIG. 16  for each run. At the end of a run, the Monte Carlo test driver collects the FTE data before terminating the simulation tool, and then the simulation tool is restarted for the next Monte Carlo run. The process is repeated until the test driver has performed the required number of runs for the Monte Carlo test. In some Examples, the simulated performance using enhanced guidance laws is compared and contrasted with simulated performance using legacy or conventional guidance laws. 
     Example Approach Plan 
     By way of another Referring now to  FIG. 17 , a federal aviation administration (FAA) specified RNP-0.11 approach plan for the KDCA (National Reagan) airport is depicted. The FAA RNP-0.11 approach plan was selected for performing the Monte Carlo tests. The DA for the approach is 491 ft. mean sea level (MSL). This approach plan was read by the Monte Carlo test driver for parsing according to the Hybrid Great Circle method  216  ( FIG. 16 ), to limit the PDE within a world-wide worst case scenario of 10 meters. The Great Circle Method-parsed flight plan is set as the active flight plan  218  to be used by the FMS. The simulated aircraft weight was set to 56,000 pounds. As shown in  FIG. 17 , there is an S-turn starting about four NM away from the DA location. 
     As seen in  FIGS. 18 and 19 , the lateral and vertical deviations for the Monte Carlo runs using the enhanced guidance laws are within their respective lateral and vertical limits. 
     As seen in  FIGS. 20 and 21 , the roll command for the FMS and the actual aircraft roll at the KDCA airport using the enhanced guidance laws are within the limits for the 100 Monte Carlo runs. The S-turn can be seen in the aircraft roll data of  FIG. 21 . Because the aircraft is gradually reducing its airspeed during the approach segment, the roll angle is not a constant value during the lateral turn. The varying roll angle is needed to compensate for the reducing airspeed for a coordinated turn with a constant turn radius. 
     Lateral Threshold 
     By way of another Referring now to  FIG. 22 , the threshold for the lateral deviation at the 95% probability is set as half of the RNP value for this Example. Therefore, with the RNP value of 0.1 NM and of world-worst PDE of 10 meters, the lateral limit is 87.6 meters (i.e., 0.5*(0.1*1852−10)). This lateral limit of 87.6 meters at the 95% probability is imposed on the approach segment  220  from the Initial Approach Fix (IAF)  222   a  or  222   b  to the Decision Altitude (DA) at location  202 . 
     Vertical Threshold 
     By way of another The vertical threshold for the vertical deviation at 99.7% probability is specified as 75 ft. (23 m) for the precision approach segment  224  from the FAF  226  to the DA  202 . In contrast, the vertical threshold for the descent and arrival phases of flight is 500 ft. (153 m). 
     LNAV Complementary Filter 
     By way of another In this Example, the filtered cross-track velocity (ctvf) time function (kt)  228  of a conventional or legacy system was plotted against the ground track to end (NM). When plotted, it was observed that the ctvf has a small offset of about 0.42 degrees with relatively small noise (e.g., ctvf fluctuates slightly due to low-frequency noise associated therewith). 
     In this Example, the CTV of a conventional or legacy system was determined using the lateral derivation (e.g., ctd_dot). When plotted, the output associated with the ctd_dot was observed to have large fluctuation due to the high frequency noise associated therewith as observed against ground track to end (NM). 
     Referring now to  FIG. 23 , using the enhanced guidance laws discussed herein, the performance of the complementary filter for level and straight flight is depicted, where lateral velocity (kt) is plotted along the y-axis and ground track to end (NM) is plotted along the x-axis. As shown in  FIG. 23 , the filtered cross-track velocity (ctvf)  228  is seen as zero mean, the cross-track velocity (CTV)  232  is seen with a small bias, and the derivative of the lateral deviation (e.g., ctd_dot)  230  is seen as zero mean with the high frequency noise discussed above. 
     Referring now to  FIG. 24 , using the enhanced guidance laws discussed herein, the performance of the complementary filter for a level flight during a flight leg sequencing from a TF leg to RF leg is depicted. The spike in ctvf on the ground track around −21.5 NM shown in  FIG. 24  is a result of the reset of the complementary filter at the beginning of a newly sequenced leg. As observed, the ctvf eventually converges after several time constants of the filter. 
     ΔBAR and Roll Angle 
     By way of another As shown in Table 1, below, three different scenarios occurred, where the aircraft was performing a lateral turn with different turn radii for a level flight at the altitude of 10,000 ft. (e.g., 3048 m), and with a Calibrated Air Speed (CAS) of about 250 knots. Design parameters used by control systems to analyze the response of a second order system are listed to compare the three different scenarios. Two of the scenarios are within two bank angle regions and one is outside two bank angle regions, where the two bank angle regions may be defined as regions where if a conventional system were used then a control system pre-roll time, ζ CS , would be inadequate for accurately performing the bank angle (e.g., region (1) ΔBAR&gt;30 degrees because pre-roll time of control systems, ζ CS , is too long, and (2) ΔBAR&lt;17 degrees, because ζ CS , is too short to compensate delay of the aircraft response). For example, in Scenario B the turn radius was about seven NM, the ΔBAR was about 10 degrees, and overshoot of the control system was about 22%; whereas, in Scenario C, the turn radius was about 11 NM, the ΔBAR was about six degrees, and the overshoot was about 31%. Because the step response of a second order system is usually preferred to be around 5%, the results of the different scenarios, such as Scenario A with approximately 5% overshoot, led to the determination of Equations (42) through (44). 
     The results of Scenario A and Scenario B, using conventional or legacy guidance laws, were plotted in multiple different charts, using roll (degrees) and lateral deviation for the y-axis and Ground Track to End (NM) for the x-axis. The results of Scenario B are depicted in  FIGS. 25 and 26 . The axes of  FIG. 25  are Ground Track to End (NM) vs. roll (deg.); and, the axes of  FIG. 26  are Ground Track to End (NM) vs. lateral deviation (meters), where control deviations  234  and integrity deviations  236  are observed in  FIG. 26 . These control deviations  234  and integrity deviations  236  are also depicted in  FIG. 28 , for Scenario B using the enhanced guidance laws (see,  FIG. 7A ). 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                   
                   
                   
                   
                   
                 Max 
                   
               
               
                   
                 Turn 
                   
                 Roll Angle 
                 Rise 
                 Settling 
                 Lateral 
                 Pre-Roll 
               
               
                   
                 Radius 
                 ΔBAR 
                 Overshoot 
                 Time  
                 Time  
                 Deviation 
                 time, 
               
               
                   
                 (NM) 
                 (deg.) 
                 (%) 
                 (s) 
                 (s) 
                 (m) 
                 ζ cs  (s) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Scenario 
                 2.9 
                 22.79 
                 4.78 
                 6.75 
                 10.25 
                 9.627 
                 6.7 
               
               
                 A 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Scenario 
                 6.6 
                 10.46 
                 22.4 
                 3.63 
                 17.75 
                 −15.01 
                 3.6 
               
               
                 B 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Scenario 
                 11.2 
                 6.2 
                 31.14 
                 2.75 
                 30.25 
                 −12.627 
                 2.55 
               
               
                 C 
               
               
                   
               
            
           
         
       
     
     By way of another In embodiments, the enhanced guidance laws are used together with the conditions of Scenarios A, B, and C. Results of using the enhanced guidance laws are depicted in Table 2, below, for comparison with the results of Table 1. Based on the overshoot and the Max Lateral Deviation parameters, it is clear that the enhanced guidance laws yield results with less overall error (e.g., less TSE), enabling increased RNP compliance. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                   
                   
                   
                   
                   
                 Max 
                   
               
               
                   
                 Turn 
                   
                 Roll Angle 
                 Rise 
                 Settling 
                 Lateral 
                 Pre-Roll 
               
               
                   
                 Radius 
                 ΔBAR 
                 Overshoot 
                 Time  
                 Time  
                 Deviation 
                 time, 
               
               
                   
                 (NM) 
                 (deg.) 
                 (%) 
                 (s) 
                 (s) 
                 (m) 
                 ζ en  (s) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Scenario 
                 2.9 
                 22.79 
                 4.78 
                 6.75 
                 10.25 
                 9.627 
                 6.7 
               
               
                 A 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Scenario 
                 6.6 
                 10.46 
                 5.6 
                 5.5 
                 15.25 
                 2.846 
                 5.45 
               
               
                 B 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Scenario 
                 11.2 
                 6.2 
                 7.15 
                 5 
                 15.38 
                 2.148 
                 5.06 
               
               
                 C 
               
               
                   
               
            
           
         
       
     
     Referring now to  FIGS. 27 and 28 , Scenario B from Table 1 and Table 2 is graphically depicted with the axes of  FIG. 27  being Ground Track to End (NM) vs. Roll (degrees), and the axes of  FIG. 28  being Ground Track to End (NM) vs. Lateral Deviation (meters). Observable in  FIGS. 27 and 28  is the increased pre-roll time associated with a pre-roll segment (e.g., segment  176  of  FIG. 7 ). Also observable in  FIG. 28  are the integrity deviations  236 , discussed above. The integrity deviations of  FIG. 28  depict RNP containment compliance (e.g., maintaining one or more thresholds). It is noted that  FIG. 27  depicts the aircraft performance for the parameters of Scenario B, using the enhanced guidance laws. 
     Referring now to  FIG. 29 , for the aircraft at a true airspeed of greater than or equal to 200 knots (e.g., VTAS  200 ), the first pre-roll time  238  for the enhanced guidance laws, ζ en , is adjusted for the two bank angle regions discussed above, while retaining the pre-roll time  240  for conventional control systems, ζ CS , outside the two bank angle regions discussed above. It is noted that in some embodiments above, the true airspeed may be less than 200 knots, in which case the second pre-roll time  242  may be weighted (e.g., more for increased speeds or less for decreased speeds) based on the airspeed, such that the first pre-roll time  238  for the enhanced guidance laws gradually becomes the second pre-roll time  242 . In some embodiments, a difference between the first pre-roll time  238  and the second pre-roll time  242  represents a recognition that for reduced aircraft speed, there may be a reduced maneuverability, necessitating a longer pre-roll time (e.g., second pre-roll time  242 ). 
     VNAV Function—Capture Mode 
     A scenario occurs in the Approach Plan consisting of the aircraft cruising at 35,000 ft. (10,668 m) at a speed of 0.8 Mach, making a three degree (3°) descent with no wind. In a conventional system, the VNAV function provides a vertical deviation of −500 ft. upon entry into the Capture Mode. In contrast, using the enhanced guidance laws disclosed herein, the VNAV function provides a 30 ft. maximum vertical deviation during the capture mode. This small amount of vertical deviation using the enhanced guidance laws is more preferable for flight displays. For example, scale factors used to display such vertical deviations may be reduced using the enhanced guidance laws. 
     Referring now to  FIG. 30 , the performance of the aircraft during the Capture Mode  194  using conventional or legacy guidance laws  244  is compared to the performance of the aircraft using enhanced guidance laws  246 . Observable in  FIG. 30  is the max 30 ft. vertical deviation as well as a gentle and desired −0.05 g maneuver. 
     Transition to Track Mode 
     Referring now to  FIG. 31 , using the conventional or legacy system, a time period associated with the Transition to Track Mode  196  is computed to be about four seconds. In contrast, the Transition to Track Mode  196  yields a time period of about seven seconds with the enhanced guidance laws. The axes of  FIG. 31  are Ground Track to End (NM) vs. Pitch Command (degrees). As depicted in  FIG. 31 , the pitch command  248   a  using enhanced guidance laws after the four second time period  250  (e.g., at the end of the Transition to Track Mode  196 ) is compared to the pitch command  252  using conventional or legacy guidance laws for the four second time period  254 . Also depicted is a pitch command  248   b  after a seven second time period  256  using enhanced guidance laws. Observable in  FIG. 31  is that the Transition to Track Mode  196  having the seven second period using the enhanced guidance laws provides the closest approximation of an ideal Transition to Track Mode, where error is minimized and the FPA at the end of the transition is close to zero, indicating an accurate tracking of the defined descent path (e.g.,  108   b ). 
     Track Mode 
     Referring now to  FIG. 32 , results are depicted on a plot of Ground Track to End (NM) vs. Actual Vertical Deviation (ft.), for legacy guidance laws  244  that determine the deviation relative to the FPA leg, and for enhanced guidance laws  246  that determine the deviation relative to the vertical arc connecting two legs (e.g., arc  190 ). The FPA discontinuity for the scenario depicted in  FIG. 32  is a discontinuity of 1.4 degrees. Observable in  FIG. 33  is that the delta pitch command (Δγ) for the enhanced guidance laws  246  starts changing approximately 0.5 NM before the delta pitch command (Δγ) for the legacy guidance laws  244 . Thus, the pitch maneuver begins earlier in using the enhanced guidance laws  246  than the legacy guidance laws  244  because the legacy laws are only determining the delta pitch command (Δγ) relative to the descent leg. In other words, the legacy guidance laws  244  do not start the pitch maneuver until the active leg is sequenced, causing an overshoot and larger vertical deviation. 
       FIGS. 32-35  are exemplary embodiments of charts depicting aircraft performance, according to the inventive concepts disclosed herein. Referring now to Table 3 below, an analysis of the data depicted in  FIGS. 32-35  was performed to determine the lower threshold for calculating the vertical deviation relative to the vertical arc during the Track Mode  198  (e.g., FPA threshold). In other words, the analysis was conducted to answer the question: ‘when does the hardware and software expend calculation processing to perform guidance relative to the arc  190  during Track Mode  198 , as opposed to performing the guidance relative to the descent leg?’ As discussed above, and as shown in Table 3, the transition point for performing the guidance relative to the vertical curve (e.g., arc  190 ) may depend on a desired experienced g-force, or how gradual the transition may be. For example, at an FPA discontinuity of 0.8 degrees, a result of a 2.63% increase in vertical deviation occurred; whereas, at a 1 degree deviation, a result of a 13.83% decrease occurred. Clearly, the more gradual transition is the 2.63% change, so in some embodiments above, the transition point is chosen to be 0.8 degrees. 
     
       
         
           
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 FPA 
                   
                   
                   
               
               
                 Discontinuity 
                 Vertical 
                 Deviation  
                 Pitch 
               
               
                 (degrees) 
                 Deviation 
                 Rate 
                 command 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 0.5 
                 Increased 19.51% 
                 Reduced 38.83% 
                 Reduced 7.29% 
               
               
                 0.8 
                 Increased 2.63% 
                 Reduced 60.94% 
                 Reduced 24.03% 
               
               
                 1 
                 Reduced 13.83% 
                 Reduced 68.68% 
                 Reduced 34.62% 
               
               
                 1.2 
                 Reduced 25.04% 
                 Reduced 73.78% 
                 Reduced 45.24% 
               
               
                 1.4 
                 Reduced 34.37% 
                 Reduced 77.37% 
                 Reduced 52.64% 
               
               
                   
               
            
           
         
       
     
     EXAMPLE—CONCLUSIONS 
     The enhanced FMS guidance laws enable the accurate tracking of the defined flight paths, including curved ground paths. The Monte Carlo simulation results against the FM specified RNP-0.1 approach (e.g., for National Reagan Airport) are presented. The results indicate that the statistical performance of the enhanced FMS guidance laws comply with the FTE requirements for RNP-0.1 flight operations. Further, lateral deviations were shown to be reduced by more than 30% on the average in affected bank angle regions. In some embodiments, a 72% reduction of lateral deviations was observed. With respect to vertical deviations, in some embodiments, vertical deviations were reduced by a multiple of ten (e.g., 10 times reduction during Capture Mode). For instance, in conventional systems, during a transition from level flight to a descent, a vertical deviation may be experienced of 250 to 500 ft. (76 to 153 m), in contrast, while using the enhanced guidance laws discussed herein, the initial, transitioning vertical deviations (e.g., during Capture Mode) were on average approximately 30 ft. (9.1 m). 
     It is to be understood that embodiments of the methods according to the inventive concepts disclosed herein may include one or more of the steps described herein. Further, such steps may be carried out in any desired order and two or more of the steps may be carried out simultaneously with one another. Two or more of the steps disclosed herein may be combined in a single step, and in some embodiments, one or more of the steps may be carried out as two or more sub-steps. Further, other steps or sub-steps may be carried in addition to, or as substitutes to one or more of the steps disclosed herein. 
     From the above description, it is clear that the inventive concepts disclosed herein are well adapted to carry out the objects and to attain the advantages mentioned herein as well as those inherent in the inventive concepts disclosed herein. While presently preferred embodiments of the inventive concepts disclosed herein have been described for purposes of this disclosure, it will be understood that numerous changes may be made which will readily suggest themselves to those skilled in the art and which are accomplished within the broad scope and coverage of the inventive concepts disclosed and claimed herein.