Feedback estimation of joint forces and joint movements Number:7,386,366 from the United States Patent and Trademark Office (PTO) owispatent Apparatus and methods are provided for estimating joint forces and moments in human beings. A forward dynamics module determines simulated kinematic data. An error correction controller forces tracking error between the simulated kinematic data and measured (or desired) kinematic data to approach zero. The error correction controller generates a modified acceleration for input into an inverse dynamics module. The estimated joint forces and moments track the measured (or desired) kinematics without the errors associated with computing higher order derivatives of noisy kinematic data.<H estimation, movements, Feedback, estima, 7386366.html, Patent, pto, 7386366

Title: Feedback estimation of joint forces and joint movements

Abstract: Apparatus and methods are provided for estimating joint forces and moments in human beings. A forward dynamics module determines simulated kinematic data. An error correction controller forces tracking error between the simulated kinematic data and measured (or desired) kinematic data to approach zero. The error correction controller generates a modified acceleration for input into an inverse dynamics module. The estimated joint forces and moments track the measured (or desired) kinematics without the errors associated with computing higher order derivatives of noisy kinematic data.

Patent Number: 7,386,366 Issued on 06/10/2008 to Dariush

Inventors: Dariush; Behzad (Sunnyvale, CA)
Assignee: Honda Giken Kogyo Kabushiki Kaisha (Tokyo, JP)

Appl. No.: 11/481,700

Filed: July 5, 2006

Related U.S. Patent Documents


Application Number	Filing Date	Patent Number	Issue Date
10151647	May., 2002	7135003
60301891	Jun., 2001
60353378	Jan., 2002

Current U.S. Class: 700/245 ; 318/560; 318/568.1; 700/189; 700/190; 700/217; 700/254; 700/258; 700/259; 700/260; 700/261; 700/262; 701/23; 901/15; 901/16; 901/21; 901/39

Field of Search: 700/189,190,217,218,245,254,258,259,260,261,262 318/560,568.1 901/15,16,21,39 701/23

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Primary Examiner: Tran; Khoi H.
Assistant Examiner: Marc; McDieunel
Attorney, Agent or Firm: Fenwick & West LLP Duell; Mark

Parent Case Text

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 10/151,647, entitled "Feedback Estimation of Joint Forces and Joint Movements", that was filed on May 16, 2002 now U.S. Pat. No. 7,135,003 and is incorporated by reference herein in its entirety. U.S. patent application Ser. No. 10/151,647 claims priority, under 35 U.S.C. .sctn. 119(e), to U.S. provisional patent application Ser. No. 60/301,891, filed on Jun. 29, 2001, entitled "A recursive, nonlinear feedback approach to estimate joint forces and joint moments from kinesiological measurements," and U.S. provisional patent application Ser. No. 60/353,378, filed on Jan. 31, 2002, entitled "Forward solutions to inverse dynamics problems: a feedback linearization approach," and both of which are incorporated by reference herein in their entireties.

Claims

What is claimed is:

1. A method for determining an estimated joint load at a joint of interest in a system, the method comprising steps of: obtaining kinematic data for the system; obtaining an input force for the system; computing a modified acceleration using at least the kinematic data; performing an inverse dynamics analysis to determine the estimated joint load at the joint of interest by recursively estimating successive joint loads until the joint of interest is reached, the inverse dynamics analysis using at least the modified acceleration and the input force; and performing a forward dynamics analysis using the estimated joint load to determine simulated kinematic data for the system.

2. The method of claim 1 wherein the kinematic data comprises center of mass coordinates and a joint angle.

3. The method of claim 1 wherein the input force comprises a ground reaction force.

4. The method of claim 1 wherein the input force comprises force plate measurement.

5. The method of claim 1 wherein the input force is equal to zero.

6. The method of claim 1 wherein the input force acts at a point of contact on a segment of the system.

7. The method of claim 1 wherein the input force comprises at least one of an internal force and an internal moment.

8. The method of claim 1 wherein the input force comprises a joint torque.

9. The method of claim 1 wherein the step of computing the modified acceleration further comprises the steps of: computing an error value representing a difference between the simulated kinematic data and the kinematic data; and applying a feedback gain to the error value.

10. The method of claim 9 wherein the feedback gain comprises at least one of a positional feedback gain and a velocity feedback gain.

11. The method of claim 9, wherein applying the feedback gain to the error value comprises: determining the feedback gain that achieves a critically damped response; and applying the feedback gain to the error value.

12. The method of claim 1 wherein the step of performing a forward dynamics analysis further comprises performing an integration to determine the simulated kinematic data.

13. The method of claim 1, wherein the step of computing the modified acceleration further uses feedback of the simulated kinematic data.

14. The method of claim 1, wherein the step of performing the inverse dynamics analysis further uses feedback of the simulated kinematic data.

15. The method of claim 1, wherein the step of performing the forward dynamics analysis further uses feedback of the simulated kinematic data.

16. The method of claim 1 further comprising the steps of: obtaining kinematic data for a second joint; computing a second modified acceleration using at least the kinematic data for the second joint; performing an inverse dynamics analysis to produce a second estimated joint load, the inverse dynamics analysis using at least the second modified acceleration and the estimated joint load; and performing a forward dynamics analysis on the second estimated joint load to obtain simulated kinematic data for the second joint.

17. A computer readable medium including program instructions for determining an estimated joint load at a joint of interest in a system, comprising: program instructions for obtaining kinematic data for the system; program instructions for obtaining an input force for the system; program instructions for computing a modified acceleration using at least the kinematic data; program instructions for performing an inverse dynamics analysis to determine the estimated joint load at the joint of interest by recursively estimating successive joint loads until the joint of interest is reached, the inverse dynamics analysis using at least the modified acceleration and the input force; and program instructions for performing a forward dynamics analysis using the estimated joint load to determine simulated kinematic data for the system.

18. The computer readable medium of claim 17 further comprising: program instructions for computing an error value representing a difference between the simulated kinematic data and the kinematic data; and program instructions for applying a feedback gain to the error value.

19. A method for determining an estimated joint load at a joint in a system, the method comprising steps of: obtaining kinematic data for the system; computing a modified acceleration using at least the kinematic data, wherein the modified acceleration is computed without using accelerations estimated from measured kinematics; performing an inverse dynamics analysis to determine the estimated joint load, wherein the inverse dynamics analysis uses at least the modified acceleration; and performing a forward dynamics analysis using the estimated joint load to determine simulated kinematic data for the system.

20. The method of claim 19, further comprising determining an input force for the system, wherein the inverse dynamics analysis further uses the input force to determine the estimated joint load.

21. The method of claim 20, wherein the input force comprises a ground reaction force.

22. The method of claim 19 wherein the step of computing the modified acceleration further comprises the steps of: computing an error value representing a difference between the simulated kinematic data and the kinematic data; and applying a feedback gain to the error value.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to human motion analysis and, more particularly, to analysis of joint forces and moments using nonlinear feedback in a forward dynamic simulation.

2. Description of Background Art

In the study of human motion, inverse dynamics analysis is conventionally used to estimate joint forces and joint moments. In a conventional inverse dynamics analysis, joint forces and joint moments are calculated from the observation of segmental movement. Inverse dynamics analysis is conventionally applied to biomechanics problems because the internal forces of human joints cannot be readily measured. Segmental movements, however, can be measured and joint angles can be inferred from the measured displacement to determine the corresponding joint forces and torques.

A problem with using inverse dynamics in the study of human motion is the error caused by calculating higher order derivatives to determine joint forces and joint moments. Methods for using inverse dynamics concepts in biomechanics are well developed if the input signals are noise-free and the dynamic model is perfect. Experimental observations, however, are imperfect and contaminated by noise. Sources of noise include the measurement device and the joint itself. Inverse dynamics methods for calculating joint moments require the calculation of higher order derivatives of the experimental observations. Specifically, the angular acceleration term is the second derivative of the joint angle and the linear acceleration is the second derivative of the center of mass acceleration. Numerical differentiation of the experimental observations amplifies the noise. The presence of high frequency noise is of considerable importance when considering the problem of calculating velocities and accelerations. The amplitude of each of the harmonics increases with its harmonic number: velocities increase linearly, and accelerations increase proportional to the square of the harmonic number. For example, second order differentiation of a signal with high frequency noise .omega. can result in a signal with frequency components of .omega..sup.2. The result of this parabolic noise amplification is erroneous joint force and joint moment calculations.

Although techniques exist for filtering the noise, filtering is difficult and time-consuming because much analysis is required to separate the true signal in the biomechanical data from the noise. For example, low-pass filtering is commonly used to reduce high frequency errors. A difficulty in low-pass filtering, however, is the selection of an optimal cutoff frequency f.sub.c. Because there is no general solution for selecting optimal filter parameters, filtering techniques often produce unreliable results.

Optimization-based approaches have been proposed to estimate joint forces and joint moments without the errors associated with performing a conventional inverse dynamics analysis. Unlike inverse dynamics, optimization-based methods do not require numerical differentiation. However, the application of optimization-based solutions is limited because the methods are computationally expensive, are not guaranteed to converge, and are generally too complex to implement.

Another problem with using inverse dynamics for analyzing human motion is that the inverse technique lacks the capability to predict the behavior of novel motions. In inverse dynamics, forces and moments are calculated from observed responses. The prediction of novel motions involves calculating the response expected from the application of forces and moments. An inverse dynamics analysis lacks predictive capability because forces and moments are calculated rather than the expected response from the application of those forces and moments.

What is therefore needed is a computationally efficient system and method that: (1) estimates joint forces and joint moments without the errors due to higher order derivatives; (2) does not require closed form, whole body analysis; and (3) is useful for predicting the behavior of human motions.

SUMMARY OF THE INVENTION

One embodiment of the present invention provides an estimation of joint loads in human beings. A joint load includes the forces and moments acting at a joint. A forward dynamics module determines kinematics through numerical integration (or simulation) of the dynamic equations of motion. An error correction controller forces a tracking error between the kinematics obtained from forward simulation and measured (or desired) kinematics to approach zero. The error correction controller generates a modified acceleration for input into an inverse dynamics module. In an embodiment, the modified acceleration represents a value determined without taking the second derivative of the measured (or desired) kinematic data. The estimated joint load, therefore, when applied to the forward dynamics module, tracks the measured (or desired) kinematics without the errors associated with computing higher order derivatives of noisy kinematic data.

In another embodiment, joint loads are recursively estimated for a planar serial link system. In a recursive method, one starts at a first end of a serial chain of segments and calculates joint loads towards a second end of the serial chain. Segments in the chain are connected together by joints and the reaction forces and moments at the joint are shared by the two connected segments. The joint loads estimated for a first segment are used in the estimation for the next segment until the joint or joints in interest are reached. That is, the output of a recursion is a force and moment calculation at the connection point for the next segment. This output is used as an input for the analysis of the next segment. A recursive method, therefore, does not require modeling the dynamics of the whole body. Although in a particular case it may be desirable to model recursively whole body dynamics, the recursive method provides flexibility that can reduce sources of error.

Recursive embodiments include open chain estimations and closed chain estimations. An open chain system is constrained with the environment at one end, and the remaining terminal segments are free. A closed chain system has more than one end in contact with the environment. The segments of the link system are numbered in order of recursion beginning with segment 1 toward segment n, which is the last segment of interest. Segment n is not necessarily the last segment in the multi-body system. Rather segment n is denoted the segment at which it is desired to stop the recursive computation such that the forces and moments of interest are found. In order to initiate the recursion, one requires the force and moment acting at the first segment. For example, in human motion analysis, the ground reaction forces under the feet are typically measured and initiate the recursion equations. The use of the ground reaction forces improves the precision of joint load estimates at the joints in proximity to the ground.

In a further embodiment, the tracking system of the present invention can be applied to closed form dynamics. The closed form system equations for an unconstrained rigid body system are described by n differential equations. Similar to the recursive embodiments described herein, a control law is used to linearize and to decouple the system dynamics.

Further features of the invention, its nature and various advantages will be more apparent from the accompanying drawings and the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate several embodiments in accordance with the present invention and, together with the description, serve to explain the principles of the invention.

FIG. 1 is an illustration of how a recursive calculation can be used to separate lower body dynamics from upper body dynamics.

FIG. 2 is a free body diagram of forces acting on segments in an open chain, planar serial link system.

FIG. 3 is a free body diagram of one segment within the serial link system.

FIG. 4 is a block diagram of a tracking system for link segment i.

FIG. 5 is a block diagram of a tracking system for link segment i illustrating further details of an error correction controller.

FIG. 6 is a flowchart illustrating a recursive tracking process.

FIG. 7 is a free body diagram illustrating a three segment, two-dimensional system.

FIGS. 8A-8C are graphs illustrating tracking accuracy for the displacement of the ankle joint of FIG. 7 using small feedback gains and ignoring accelerations.

FIGS. 9A-9C are graphs illustrating tracking accuracy for the displacement of the knee joint of FIG. 7 using small feedback gains and ignoring accelerations.

FIGS. 10A-10C are graphs illustrating tracking accuracy for the forces and moments of the knee joint of FIG. 7 using small feedback gains and ignoring accelerations.

FIGS. 11A-11C are graphs illustrating tracking accuracy for the forces and moments of the hip joint of FIG. 7 using small feedback gains and ignoring accelerations.

FIGS. 12A-12C are graphs illustrating tracking accuracy for the displacement of the ankle joint of FIG. 7 using small feedback gains and including accelerations.

FIGS. 13A-13C are graphs illustrating tracking accuracy for the displacement of the knee joint of FIG. 7 using small feedback gains and including accelerations.

FIGS. 14A-14C are graphs illustrating tracking accuracy for the forces and moments of the knee joint of FIG. 7 using small feedback gains and including accelerations.

FIGS. 15A-15C are graphs illustrating tracking accuracy for the forces and moments of the hip joint of FIG. 7 using small feedback gains and including accelerations.

FIGS. 16A-16C are graphs illustrating tracking accuracy for the displacement of the ankle joint of FIG. 7 using large feedback gains and ignoring accelerations.

FIGS. 17A-17C are graphs illustrating tracking accuracy for the displacement of the knee joint of FIG. 7 using large feedback gains and ignoring accelerations.

FIGS. 18A-18C are graphs illustrating tracking accuracy for the forces and moments of the knee joint of FIG. 7 using large feedback gains and ignoring accelerations.

FIGS. 19A-19C are graphs illustrating tracking accuracy for the forces and moments of the hip joint of FIG. 7 using large feedback gains and ignoring accelerations.

FIGS. 20A-20C are graphs illustrating tracking accuracy for the displacement of the ankle joint of FIG. 7 using large feedback gains and including accelerations.

FIGS. 21A-21C are graphs illustrating tracking accuracy for the displacement of the knee joint of FIG. 7 using large feedback gains and including accelerations.

FIGS. 22A-22C are graphs illustrating tracking accuracy for the forces and moments of the knee joint of FIG. 7 using large feedback gains and including accelerations.

FIGS. 23A-23C are graphs illustrating tracking accuracy for the forces and moments of the hip joint of FIG. 7 using large feedback gains and including accelerations.

FIG. 24 is a graph illustrating tracking error for the displacement of the ankle joint of FIG. 7.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Preferred embodiments of the present invention are now described with reference to the accompanying figures where like reference numbers indicate identical or functionally similar elements. Also in the figures, the left most digit of each reference number corresponds to the figure in which the reference number is first used.

FIG. 1 is an illustration showing how recursive calculation is used to separate lower body dynamics from upper body dynamics. The illustration includes upper body portion 105 and lower body portion 110. A segment of upper body portion 105 is illustrated with load 140. Lower body portion 110 includes segments having ankle joint 120, knee joint 125, and hip joint 130. In a recursive method for computing joint forces and moments, the upper body portion 105 can be modeled separately from the lower body portion 110. Starting with ground reaction forces 115, one can effectively isolate joints 120, 125, and 130 of lower body portion 110 as well as the associated link segment parameters, from upper body portion 105. That is, the internal forces and moments acting on joints 120, 125, and 130 can be estimated without considering the effects due to load 140 or the physical parameters of upper body portion 105, such as mass, center of mass, inertia, and segment length. These parametric uncertainties in upper body portion 105 are significant sources of error in the estimation of the internal forces and moments in the human body when using closed form, whole body dynamic procedures. In contrast to a closed form whole body solution, embodiments of a recursive solution use measurements of ground reaction forces 115 as constraints to compute recursively the joint moments from the ground and working up toward the, e.g., knee joint 125 and hip joint 130.

An advantage of using a recursive method to estimate joint forces and moments is the ability to focus on joints of interest without introducing additional sources of error. Further, a recursive method with measurements of ground reaction forces 115 as input, for example, provides an extra sensing modality. That is, combining kinematic and reaction force information provides an additional cue that increases the reliability of the resulting internal force estimations. Human beings may be subjected to unpredictable loads, or constrained dynamics resulting from interaction with other objects or other people in the environment. Such circumstances can alter the dynamic representation required to estimate the internal forces and moments at the joints. Some applications for using a recursive method in these circumstances include performing biomechanical studies of lifting tasks and developing controls for assistive devices that aid the physically impaired in everyday tasks. One skilled in the art will appreciate that in-shoe force and pressure sensing devices are complementary technologies that can be used to provide extra sensing modalities for use in various force and moment estimations.

Recursive Method for a Two-Dimensional Serial Chain System

Embodiments of the present invention are applicable to planar systems including recursive open chain and recursive closed chain. In an open chain system, at most one end of the multi-body system is in contact with the environment. The other end or ends are free or unconstrained. In a closed chain system, more than one end is in contact with the environment.

FIG. 2 is a free body diagram of forces acting on segments in an open chain, planar serial link system. The system includes first segment 205, second segment 210, and n.sup.th segment 215. The segments 205, 210, and 215 are linked by revolute joints. Each of segments 205, 210, and 215 is illustrated as a free body diagram in which first joint 220, second joint 222, third joint 224, and n.sup.th joint 226 connect the segments. First segment 205 includes first joint 220 and second joint 222. Second segment 210 includes second joint 222 and third joint 224. In particular, segments 205 and 210 are connected as follows: second joint 222 links first segment 205 with second segment 210. Therefore, a serial chain of n segments is formed by connecting the n segments at the common or overlapping joint.

For each of joints 220, 222, 224, and 226, the joint torques, the horizontal reaction forces, and the vertical reaction forces at each joint is illustrated and denoted by .tau..sub.i, F.sub.i, and G.sub.i respectively. For example, with respect to first joint 220, the joint torque .tau..sub.1, horizontal reaction force F.sub.1, and vertical reaction force G.sub.1 are illustrated. With reference to FIG. 2, an instance of a recursive calculation is now described. In a recursive calculation, a multi-body system is conceptually separated into individual segments. The free body diagram of each segment is analyzed. The segments are connected together by joints, e.g., second joint 222. The reaction forces and moments at, e.g., second joint 222 are shared by first segment 205 and second segment 210. Analysis begins at first segment 205, where the force and moment at the connection point of the second segment 210, i.e., second joint 222 are computed. The calculated forces and moment at second joint 222 are the output of recursion 1. This output is used as input for the analysis of the next segment, e.g., second segment 210. The recursive analysis of segments continues until n.sup.th segment 215 is reached. The n.sup.th segment 215 is the segment of interest or the segment at which it is desired to stop the recursive computation. In an embodiment where ground reaction forces 115 (FIG. 1) are acting on first segment 205, one computes the forces and moment acting on second joint 222 in terms of the forces and moment acting on first joint 220. Next, one computes the forces and moment acting on third joint 224 in terms of the previously computed forces and moment acting on second joint 222. This recursive procedure of using the output of dynamics calculations as input for the next calculation is repeated until forces and moments have been found for the joint or joints in interest. One skilled in the art will appreciate that n.sup.th segment 215 is not necessarily the last segment in the multi-body system. Rather, n.sup.th segment 215 is denoted the segment at which it is desired to stop the recursive computation such that the forces and moments of interest are found. It should further be noted that ground reaction forces 115 act at the point of contact, which is not necessarily at the joint. Further details of the calculations are described below and with respect to FIG. 3.

FIG. 3 is a free body diagram of one segment within the serial link system. Body segment 305 represents the i.sup.th segment of a planar serial link system, such as the system illustrated in FIG. 2. Body segment i includes joint i (310) and joint i+1 (315). For isolated body segment i, where i=1 . . . n, the acceleration of the center of mass is ({umlaut over (x)}.sub.i, .sub.i), the joint angle with respect to the vertical is .theta..sub.i, and the angular acceleration is {umlaut over (.theta.)}.sub.i. As illustrated in FIG. 3, the physical parameters for body segment i are mass m.sub.i, moment of inertia I.sub.i, segment length l.sub.i, and length to center of mass k.sub.i. Also illustrated in FIG. 3 for each of joints 310 and 315 are joint torques .tau..sub.i, horizontal reaction forces F.sub.i, and vertical reaction forces G.sub.i. The Newton-Euler equations for computing the forces and moments at each of joints 310 and 315 of body segment i are set forth below as Equation 1, Equation 2, and Equation 3. m.sub.i{umlaut over (x)}.sub.i=F.sub.i-F.sub.i+1 (1) m.sub.i .sub.i=G.sub.i-G.sub.i+1-m.sub.ig (2) I.sub.i{umlaut over (.theta.)}.sub.i=-F.sub.ik.sub.i cos(.theta..sub.i)+G.sub.ik.sub.i sin(.theta..sub.i)-F.sub.i+1(l.sub.i-k.sub.i)cos(.theta..sub.i)+G.sub.i+1- (l.sub.i-k.sub.i)sin(.theta..sub.i)+.tau..sub.i-.tau..sub.i+1 (3)

One skilled in the art will appreciate that Equation 1 represents an expression for summing the forces acting on body segment 305 in the x or horizontal direction. Similarly, Equation 2 represents an expression for summing the forces acting on body segment 305 in the y or vertical direction. In Equation 2, the gravitational acceleration is represented by g. Equation 3 represents an expression for summing the angular accelerations acting at joints 310 and 315.

Inverse Dynamics Methodology

In inverse dynamics analysis, the forces and moments acting at the joints are computed from measured or desired kinematic data. Kinematic data includes center of mass coordinates and joint angle data. In an embodiment of the present invention, a recursive solution to compute the forces and moments at each joint can be obtained from a compact representation (matrix form) of the Newton-Euler equations as set forth in Equation 4 below. In Equation 4, U.sub.i=[F.sub.i G.sub.i .tau..sub.i].sup.T is a vector (transposed) whose elements correspond to the horizontal force, vertical force, and moment acting at joint i (310), respectively. The forces and moment at joint i+1 (315) are described by U.sub.i+1. Further details of a recursive solution for U.sub.i or U.sub.i+1 are described below. M.sub.i{umlaut over (q)}.sub.i=A.sub.iU.sub.i+1+B.sub.iU.sub.i+P.sub.i (4)

Vector q.sub.i=[x.sub.i y.sub.i .theta..sub.i].sup.T represents the center of mass coordinates and joint angle at joint i. One skilled in the art will appreciate that the term {umlaut over (q)}.sub.i in Equation 4 represents the second derivative of vector q.sub.i. The elements of Equation 4 are defined in additional detail as follows:

.times..times..theta..times..times..times. ##EQU00001## .times..function..theta..times..function..theta..times..times..tau. ##EQU00001.2## .times..function..theta..times..function..theta..times..times..tau. ##EQU00001.3##

Open Chain Estimation

As described above, an open chain system has one end in contact with the environment. The one end in contact with the environment is termed a constrained end. In an embodiment of the present invention, the constrained end is a human being's feet that are in contact with the ground or other supporting surface. In one embodiment, kinematic data is supplemented with measurements of ground reaction forces 115 (denoted U.sub.1) to improve the precision of the internal force and moment estimations. The segments are numbered from the "ground up" from 1 to n, with n being the last segment of interest. Thus the Newton-Euler inverse dynamics methodology makes use of measurements of the forces and moments under the feet. With U.sub.1 available as a boundary condition at segment 1, the force and moment at segment i, where i has a value from 1 to n (i:1.fwdarw.n) is computed successively with Equation 5, starting with segment 1 and working toward segment n. In an open chain estimation and with n being the last segment in the chain, there are no external forces applied at segment n, such that U.sub.n+1=0. U.sub.i+1=A.sub.i.sup.-1[M.sub.i{umlaut over (q)}.sub.i-B.sub.iU.sub.i-P.sub.i] (5)

Due to noisy measurements and errors in the biomechanical model, the boundary condition at the segment that is free is generally violated. In other words, in an open chain embodiment whereby the recursion proceeds from segment 1 to the free segment (denoted by n), then U.sub.n+1 is not equal to 0. The over-determinacy is resolved by adding residual forces and torques to segment n. An advantage of a recursive formulation, numbering the segments from 1 to n, is that the entire body need not be modeled. The force and moment estimation is complete at segment n regardless of whether segment n is the last segment of the serial system. The parametric uncertainties in the upper extremities and uncertainties in the rigid-body model are significant sources of error in the estimation of the internal forces and moments. These uncertainties in the upper extremities can be avoided, however, when only joint moments proximal to the force plate are desired.

In another open chain embodiment, only kinematic measurements are available. The segments are labeled from 1 to n, wherein segment 1 has a free end, rather than a constrained end. Because segment 1 has a free end, U.sub.1=0 is available as a boundary condition for the recursion to segment n.

Closed Chain Estimation

In another embodiment of the present invention, closed chain estimation is performed. As described above, a closed chain system has more than one end in contact with the environment. In this embodiment, sensor measurements or other initial forces are needed to estimate the internal forces and moments. The segments are serially numbered beginning with segment 1 toward segment n, which is the last segment of interest. The sensor measurement or initial force at segment 1 is denoted U.sub.1, wherein U.sub.1.noteq.0 because the end of segment 1 is constrained. With measurements for U.sub.1 available as a boundary condition at segment 1, the force and moment at segment i, where i has a value from 1 to n (i:1.fwdarw.n) is computed successively with Equation 5, starting with segment 1 and working toward segment n.

Inverse Solution Using Nonlinear Feedback

FIG. 4 is a block diagram of a tracking system for body segment i. Error correction controller 405, inverse dynamics module 410, and forward dynamics module 415 are coupled to form a tracking system. Inputs to error correction controller 405 include kinematic data q.sub.m.sub.i, {circumflex over ({dot over (q)}.sub.m.sub.i, and {circumflex over ({umlaut over (q)}.sub.m.sub.i, as well as s