Modeling of control and learning in a stepping motion

Biol. Cybern. 55, 387-396 (1987) Biological Cybernetics @ Springer-Verlag 1987

Modeling of Control and Learning in a Stepping Motion

H. Flashner 1, A. Beuter 2, and A. Arabyan 1

1 University of Southern California, Los Angeles, Department of Mechanical Engineering, Los Angeles, CA 90089, USA 2 Universit6 du Qu6bec fi Montr6al, Canada

Abstract. In a previous study (Beuter et al. 1986) the authors modeled a stepping motion using a three-body linkage with four degrees of freedom. Stepping was simulated by using three task parameters (i.e., step height, length, and duration) and sinusoidal joint angular velocity profiles. The results supported the concept of a hierarchical control structure with open- loop control during normal operation. In this study we refine the dynamic model and improve the simulation technique by incorporating the dynamics of the leg after landing, adding a foot segment to the model, and preprogramming the complete step motion using cycloids. The equations of the forces and torques developed on the ground by the foot during the landing phase are derived using the Lagrangian method. Simulation results are compared to experimental data collected on a subject stepping four times over an obstacle using a Selspot motion analysis system. A hierarchical control model that incorporates a learning process is proposed. The model allows an efficient combination of open and closed loop control strategies and involves hardwired movement segments. We also test the hypothesis of cycloidal velocity profiles in the joint programs against experimental data using a novel curve-fitting procedure based on analytical rather than numerical differentiation. The results suggest multiob- jective optimization of the joint's motion. The control and learning model proposed here will help the understanding of the mechanisms responsible for assembling selected movement segments into goal- directed movement sequences in humans.

1 Introduction

One of the difficulties encountered in the study of movement control is to separate the expression of mechanical demands of the task from the behavior of

the nervous system. This task is complicated by the fact that in highly learned skills, such as stepping, the body structures its patterns of movement to minimize various performance objectives such as jerk, energy, torque, etc. (Nelson 1983) whose combination is not well understood (Marshall et al. 1984). However, the knowl- edge of the objectives optimized by the central nervous system (the performance criterion) should help us understand the coordination and regulation of human movements (Zajac 1984). In an effort to understand how the nervous system simplifies the organization of motor acts and their control to a manageable task, it is important to develop physiologically relevant models and to rely more heavily on simulation techniques (Zajac 1984).

It is equally important to develop these models in such a way that movement learning, adaptation and generalization are integral parts of the model. Raibert (1978) proposed a model in which the equations of motion of a limb are expressed in a parametric form (i.e., the equations are restated with a subset of the independent variables held constant) that facilitates transformation of desired trajectories into motor plans. Such parametric equations are used in conjunc- tion with what Raibert calls Quantized Multidimen- sional Memory (QMM) organized by state variables. QMM are supplied with data produced from practice movements. Raibert's model is powerful and simple, and it integrates the ability to learn new movements. However, the organization of QMM in which space is divided into segments and entire matrices are stored does not appear to be a realistic approach to learning since for each movement an entire table of numbers must be stored. In addition, this model appears unable to handle unexpected changes in mechanical loads.

In a previous study, Beuter et al. (1986) used a phase plane approach to develop a four degree of freedom model of leg motion in a stepping taks. Optimal simulation results for a three link leg model

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were obtained by using three task parameters (i.e., step height, step length, and step duration), sinusoidal velocity profiles for the hip and knee joints, and inverse kinematics for determining the extreme values of the hip and knee joint angles. The results supported the concept of a hierarchical control logic to accommodate errors and compensate for unexpected events. The results also suggested the existence of simple preset programs driving the joints to produce the desired motion over the obstacle and requiring a minimum of computation time. This model, however, included neither a foot segment, nor the dynamics of the leg after landing. In the present paper we address these limita- tions, and propose a more efficient hierarchical structure and learning procedure, while keeping in mind that the improved model should remain physiologically relevant and computationally simple.

2 Experimental Procedure

Light emitting diodes (LEDs) were secured on the head of the fifth metatarsal, lateral malleolus, lateral knee condyle, greater trochanter, and acromion process of a nonhandicapped adult female. The five LEDs were connected by flexible cables to the LED control unit carried on a belt by the subject. Two infrared cameras placed at a 60 degree angle were used to measure the position of the LEDs in the right sagittal plane after the X-Y-Z coordinates of the reference LEDs were calibrated. A Selspot system (SELCOM) interfaced with a PDP-11/23 was used to collect kinematic data at a sampling rate of 277 Hz. The subject was asked to step four times over a low obstacle at a natural cadence starting and finishing from a standing position. The ankle, knee and hip joint angles were calculated from the X and Y coordinates of the LEDs measured from a fixed frame of reference. For a more complete descrip- tion of the experimental methods used see Beuter and Cowden (1986). For the purposes of this paper only the raw experimental data collected was used in the development of the model (see Fig. 1).

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Time (sec.)

Fig. 1. Position of the hip and knee angles as a function of time in the stepping motion performed by the subject

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3 Sys tem Mode l

The flow of information diagram of a stepping motion model is shown in Fig. 2. As in any model of a controlled system the proposed model consists of two parts: system dynamic model (the plant) and motion control logic (control law). The system's characteristics are described in terms of differential and algebraic equations that give a functional relationship between the torques and forces (commanded variable) acting at the various joints and the resulting trajectory (output variable). The control logic measures actual trajectory data and generates the required control torques needed to complete the task.

3.1 Dynamic Model

This part of the system's model can be divided into two sets of functional relationships:

(i) Dynamic equations, which relate the generalized coordinates used to describe the motion, such as joint angles and joint translations, to the control torques and forces.

(ii) Kinematic equations, which relate the motion of various parts of the leg, for example the heel, to the generalized coordinates.

3.1.1 Dynamic Equations. The dynamic equations are given by a set of n nonlinear second order differential equations. Since the stepping motion has two phases, the swing phase and the landing phase, with different number of degrees of freedom, it is convenient to describe the motion with the aid of Lagrange multipliers that represent the ground forces FGx and FGy as shown in Fig. 3. By using this formulation, the equations remain in the same form for both phases of motion as follows:

M(0)0 + f(0, 0) = Q. (1)

of the formulation see Part B of the For details Appendix.

DISTURBANCES

l LEG DYNAMICS

LEO TRAJECTORY (OUTPUT)

T ONTROL COMMANDS (TORQUES)

TAgK (STEPPING)

CONTROL LOGIC

F Fig. 2. Information flow diagram of the system model

TRAJECTORY PARAMETERS

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@ Begin swing

@ Mid swing

@ Begin londing

@ Mid Ionding

@ End Ionding

Fig. 3. Schematic representation of the leg dynamic model

| | | | IS I , m I TH

3.1.2 Kinematic Equations. The kinematic relationships for the leg consist of a set of algebraic equations that describe the motion of any point on the leg in terms of the joint coordinates that are directly affected by the control torques. The kinematic equations can be

written as:

x,=R(0), (2) where x, is the trajectory of a given point.

When any of the kinematic relationships are prescribed, the kinematic equations become constraint equations accompanied by a set of constraint forces (Lagrange multipliers).

3.2 Motion Control Logic

This part models the behavior of the neuromuscular system controlling the motion. As stated previously (Beuter et al. 1986) the control logic model consists of up to four basic modules. These modules are described below and illustrated in Fig. 4.

3.2.1 Trajectory Generator. This part of the motion control logic computes the trajectory that a point P on the leg must follow to achieve the goals of the task. In the particular case of a stepping motion, the point P can be a point on the foot or the hip joint. It is obvious that a large number of possible trajectories can achieve the goals of the task and that there are many ways to step over the obstacle. The path chosen, and the required control commands required to generate it, are computed by the forward cbntrol logic that selects the optimal trajectory by minimizing a predetermined performance criterion.

3.2.2 Forward Control Logic. Given a set of possible trajectories that can perform the task at hand, this part of the control logic selects the best trajectory and generates the required forces and torques. The performance of a trajectory is determined by the value of a performance criterion. Minimizing time, energy, peak

TASK b I

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TRAJECTORY

GENERATOR

I

PR[PLANNED CONTROL UNIT

TRAJECTORY Xp(t)

-I

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FORWARD NOMINAL TORQUE 5 APPL~D TORQUE (Q)

CONTROL LOGIC I ~I+ -

. . . . J I I c~176 T~ 4'')

INTERACTIYE UNIT

FEEDBACK

CONTROL LOGIC

t i SENSING

LOGIC v TRLJECTORY PARAMETERS

_ _ _ /

Fig. 4. Control logic diagram

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force, or any combination of the above are possible performance criteria. The differential and algebraic equations given by (1) and (2) serve as constraints. Additional constraints can be imposed on the system. Possible constraints can be limits on the contact force, velocities, accelerations, etc. Mathematically the optimization problem is given by:

Minimize J = P(x,, Q, 0) (3)

subject to:

(i) h(O, Q) >0 imposed contraints (4)

(ii) 0 = R - l(x,) kinematic constraints (5)

(iii) Q = M(0)O + f(0, 0) dynamic constraints. (6)

3.2.3 Sensing Logic. This part models the mechanism that detects and reports to the CNS about the status of motion. The sensed signals can be joint angles, position of a point on the leg, e.g. foot, or contact forces with the ground, etc. It should be noted that by the use of Lagrange multipliers measuring the foot's motion becomes equivalent to measuring the ground reaction forces. We can either measure the motion xp and then k is unknown and must be calculated, or we can measure k and then the motion xp is computed. It should also be noted that the sensing is not done continuously. The measurement of the trajectory parameters is done at prescribed intervals of time T~ (sensing sampling time). The sampling time need not to be constant. Typically the sampling time is shorter at more critical phases of motion, i.e. when sudden changes in the trajectory take place. Thus an accurate model of the control logic results in a discrete time system with nonuniform sampling period (see Jury 1958).

3.2.4 Feedback Control Logic. This part models the corrective measures (correction torques, forces, changes in precomputed trajectory, etc.) that must be implemented to complete the required task. Correction commands are applied when the system does not follow the precomputed trajectory as calculated by the feedforward logic. The performance of the task can be specified in terms of a point path in space, and/or in terms of the expected forces during motion. The forward control unit expects the foot to follow a certain path, land at a certain location, and expects a force of certain magnitude when the landing part of the stepping motion begins. If any of these variables are not within the required range, a corrective signal is issued by the feedback logic. Since the feedback control signal is applied only when needed, the activity can be represented as a sampled data system with sampling time T~. In general T~ is longer than T~ and is determined by the control logic.

4 Leg's Equations of Motion

The mechanical model of the leg is shown in Fig. 3. The equations of motion for the swing phase were derived in the previous paper (Beuter et al. 1986). As explained in Sect. 5, the contact with the ground during landing introduces constraints which reduce the number of degrees of freedom by two. We model the constraints by introducing Lagrange multipliers which represent the constraint forces. This formulation maintains the equations in the same form for both the swing and landing phases of the stepping motion. The equations of motion for the swing phase are given in the Appendix [(22)-(25)1. The expressions for the hori- zontal and vertical components of the ground force on the foot as well as the torques developed at the joints are given in Eqs. (31)-(34) of the Appendix. A model of a foot has been added to the overall model and is assumed to be a massless rigid link attached to the extremity of the shank. Under this assumption the foot plays no role in the swing phase of the motion. Hence, the equations of motion for the swing phase remain unchanged. The role of foot during landing is discussed below,

Figure 5 shows the ground forces and joint torques during the landing phase when cycloidal velocity profiles are programmed into the joints.

The addition of the foot as a weightless rigid link has only the effect of moving the point of action of the ground force along its length while the leg completes

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Fig. 5. Forces and torques for pure cycloidal velocity profiles

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the landing phase of the motion. This is because during the landing phase the foot pivots around the ankle and therefore remains in contact with the ground along its entire length. Thus, the addition of the foot does not change the expressions for FGx and F~y, but introduces an additional torque which is given by

Tp: {J3[- + 0",3 cos(0. + 0,3 + (0. + 0,3 2

x sin(0n + OK)] + J4(0"f/cos On-- 02 sin O~t)}xa, (7)

where x~ is the distance of the point of action of the ground force from the ankle joint (see Fig. 3). Ex- pressions for J3 and Jr are given in the Appendix. It is assumed for the purposes of this paper that the ground force moves along the foot linearly in time, in the following way:

xG=lr t>tl , (8)

where 14 is the length of the foot, tl is the time landing begins, and t 2 is the time landing ends.

The effect of the foot on the total torque on the leg (Tn + TK) is shown in Fig. 5. As can be seen, the effect of the foot is a minor reduction in positive torque. Therefore, its presence has little importance in the case of stepping motion. It may play a more prominent role,

LEVEL 3 DECISION UNIT

TASK STATUS PERFORM ABORT TASK

LEVEL 2

LEVEL I

LEVEL 0

-I I NTERACTI VE

CONTROL UNI T 'T ACTUAL TRAJECTORY

I TASK PARAMETERS (h)

PREPLANNED

CONTROL UNIT

TRAJECTORY I PARAMETERS

DISTURBANCES j

LEG DYNAMICS

NOMINAL TORQUES Q*

L Fig. 6. Hierarchical control model for the stepping motion

however, when it has to act as an active control device under the interactive control unit in the control hierarchy (see Fig. 6).

5 Control Law Structure

It was shown in a previous paper (Beuter et al. 1986) that the stepping motion is a preplanned operation learned and stored in the nervous system. The preplanned motion model was obtained by applying phase plane analysis to the swing portion of the stepping motion. In the previous work it was also suggested that the preplanned trajectories are defined by a small number of parameters. In this section, the preplanned motion model is extended to include both the swing and landing phases of the stepping motion. It is shown that the preplanned motion model is a natural result of the hierarchical structure of the control logic proposed for stepping (see Beuter et al. 1986).

As discussed in Sect. 3 control logic for stepping allows both open and closed loop control operations. An efficient control strategy requires, therefore, coordination among the various subsystems, learning of the optimal open loop control commands, definition of alternative operation modes when unexpected events occur, and compensation for interactions between the subsystems. A hierarchical organization of the control logic allows the performance of all these activities.

A hierarchical structure of stepping motion is shown in Fig. 6. The highest level of the hierarchy decides about the performance of the task (stepping) and gives the command to abort the task or change the strategies of the forward (preplanned) and feedback (interactive) control units. During normal operation the interactive control unit is activated only when corrections to the preplanned command are needed. However, since this unit possesses a sensing mechanism that can evaluate the performance of the task, during the learning stage this level is used continuously for evaluation of given trajectories. In other words during the learning stage the interactive unit solves the optimization problem defined by (3)-(6). The learning process is explained as follows: The trajectory generator sends a candidate trajectory Xp(t) to the forward control subsystem. Using (5) and (6) the joint angles and the control torques are calculated. In addition, the performance index is computed and stored in memory. During learning, the task is repeated many times with different candidate trajectories yielding different values for the performance index. Comparison of performance indices of the various trajectories yields the best control strategy for Q.

The hierarchical control structure allows additional sophistication of the optimization process. It is clear that stepping over obstacles with different

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heights does not require learning for each particular obstacle as is required by Raibert's model (1978). Experimental data (see Beuter et al. 1986) shows that the trajectories for stepping over different obstacles are consistent. The limited variability observed in the trajectories is found in the magnitude of some parameters (peak height, landing location etc.). The hierarchical organization of the control allows performance of stepping in different conditions without repeating the optimization procedure. The optimal trajectory x v is parameterized by relevant variables of motion such as step height, step length, and step duration. After the completion of the learning process a family of optimal trajectories is stored in the preplanned control unit. The most efficient way to do this is to parameterize the optimal strategy using a limited number of parameters. Following this logic, an optimal trajectory xp yields optimal joint angles in the form of a finite series:

0 =~, c, (h)f~ (h, t), (9)

where h is the vector of relevant task parameters. Similarly using (6) the corresponding optimal

torques can be also expressed in terms of a finite series:

Q = ~2 d,(h)9~(h, t). (10)

The preplanned unit stores the generic functions f~(h, t) and the functional relationships c~(h). Alterna- tively, di and g~ of (10) can be stored. The number of base functions, and therefore the number coefficients, needed to generate a trajectory is usually low. As shown in Sect. 6, in the case of the stepping motion three parameters and an appropriate choice of base functions f~(h, t) generate a feasible trajectory. The functional relationships ei(h) are also downloaded and stored. However, as shown in Sect. 6, the functions fi(h, t) originate from a set of predetermined functions that are stored in the nervous system before the optimization process begins. This type of procedure implies that the functional optimization problem defined in (3), i.e., finding a function e(t) or alternatively a function Q(t), is replaced by a simpler problem of finite dimensional optimization. The optimization problem (3) through (6) can be reformulated as:

Min Min {P(0, Q, xp)} (11) h f E F

subject to kinematic and dynamic constraints and where 0 is given by (9). F is a finite set of known functions:

V = {f~(h, t), i= 1, 2,..., k}. (12)

The set of parameters e~(h) varies for various phases of motion. When the leg moves in the swing mode one

set of parameters ci(h) is utilized. Once the landing phase starts a new set of parameters is employed. The starting of the landing phase is communicated to the preplanned control unit by the sensing logic of the interactive control unit (see Figs. 4 and 6). When a constraint force 2 of contact between a foot and the ground is detected a change of operational modes is commanded. The change command is simply a command to utilize a new set of parameters ci(h).

The hierarchical structure and the learning procedure proposed here allow efficient combination of open and closed loop control logics. Open loop control has the advantage of fast action but lacks the ability to compensate for the effects of disturbances. Closed loop control is ideal to counteract the effects of disturbances but involves complex activities that require time, and memory storage. Combining both control logics in a hierarchical structure with the bulk of the activity performed open loop by the preplanned unit, and delegating the closed loop activity to a higher hierarchy allows both fast operation and compensation for disturbances.

The learning process explains the inherent motion abilities of new born babies by postulating the existence of hardwired movement segments in the form of basis functions f~(h, t). In addition the proposed model explains the learning process of every individual by suggesting that this part of learning involves the

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Fig. 7. Position of the hip and knee angles as a function of time in a stepping motion performed by a 3-week old infant supported under the arms

selection of undetermined parameters ci(h). Piaget (1952) has hypothesized that complex skills begin in a matrix of unlearned movement segments (Adams 1984). These segments appear to be the necessary building blocks for more complex movement sequences to develop. Andre-Thomas and Autgaerden (1966) have shown that a few minutes after birth an infant can step over an obstacle without stumbling. Data on a stepping motion performed by a three-week old infant supported under the arms is presented in Fig. 7. As can be seen on Fig. 7, the hip and knee joint angular displacements ate similar to those performed by an adult (see Fig. 1). These movement segments or building blocks are available at birth, at a time when the cortical structures are very poorly developed especially in the motor areas associated with the legs (Thelen 1981). This seems to indicate that infants use an open loop control logic in limb movements.

6 Preprogramming and Curve Fitting

6.1 Preprogramming

In a previous paper (Beuter et al. 1986), the hypothesis of a preprogrammed stepping motion was proposed and its feasibility was demonstrated using sinusoidal velocity profiles for the joint programs. Here, we refine the joint programs by assuming cycloidal velcotiy profiles.

6.1.1 Hypothesis of Cycloidal Motion. Cycloidal velocity profiles for the hip and knee joints are given by:

O.=c (co-co coscoO ,

0K = c2(co- co cosco0,

03) (14)

where cl, c 2, and co are constants to be determined by the taks parameters.

The motivation for selecting cycloidal velocity profiles is that unlike in sinusoids, both the first and second time derivatives vanish at the beginning (cot = 0) and at the end (cot = 2n) of the motion. This is more in agreement with experimental data and com- mon experience because we know that at the start and end of a stepping motion both the velocity and acceleration at all the joints are zero. This has the effect of smoothing the transition from a stationary state to a moving state and from a positive velocity to a negative velocity. Furthermore, the introduction of cycloids does not require new task parameters to describe the motion of the joints, since cycloids are described by the same number of parameters as sinusoids.

As done in the previous paper, we integrate (13) and (14) and rewrite the constants c t and c2 in terms of the

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hip and joint constants to be computed, and we obtain:

iOH2[4nt _ s i n ~ t ) 1 2nL t,

O<t<tj2 (0 m -- 0It2) [ (4n( t - - t j 2 )

2re " ( t j2 < t < t~,

O<t<=tj2

OK=(n--OK2) I4n ( t - - t j 2 ) - - s in f4n ( t~ t j2 ) ] }2n ( ~ (16)

t j2 < t <= t,,

(15)

where 0nz and 0KE are, respectively, the hip and knee joint angle displacements corresponding to the max- imum raised position of the foot and 0m is the hip joint angle displacement corresponding to the stage when the foot touches the ground again (see Fig. 8). The same assumptions made for the sinusoidal case are made here and the variables Ore, 0n2, and 0K2 are determined using the equations in Part B of the Appendix.

6.1.2 Correlation with Experimental Data. A novel curve-fitting method was used to test the hypothesis of cycloidal velocity profiles in the joint programs against experimental data. An advantage of the method is that it eliminates the need for numerical differentiation to compute first and second time derivatives of the variables measured and all the problems associated with numerical differentiation.

Path of hip joint ml#l% Poth of foot

Obstecle

OK

OH 0m - ~ ~ 1

Time tr

Time tr

JOINT PROGRAMS

Fig. 8, Path of the foot generated by cycloidal motion at hip and knee joints

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180

140 ~

120

100 2.0

I I I

0,0 0.5 i.O 1,5 Time (sec)

- - Experimental data . . . . Perfect cycloid . . . . . . . SVD fitted data

Fig. 9. Comparison of experimental data and various curve fitting approaches

In general, the data to which the curve is to be fitted is assumed to be a function of the form:

G(t) = elf ,( t) + Czfz(t) +. . . + cmf,,(t), (17)

where the f l (i= 1,2, ..., m) are arbitrarily selected, linearly-independent functions known as basis functions or the basis, t is the independent variable (in our case t is the time but in general it can be any independent variable) and the ci (i= 1,2, ...,m) are parameters to be determined. Thus the curve-fitting problem becomes one of selecting basis functions f~ and parameters ci such that we get the least error between the original data and points on G(t). For a given set of f~ the problem becomes a linear least- squares approximation problem which is solved using a FORTRAN subroutine.

Once the ci are found their relative magnitudes act as measures of the importance or the dominance of the corresponding fi in G(t), and depending on those results one may judiciously drop some of the components in G(t) as "noise terms" and thus smooth out the data.

Because this procedure gives the user the option of selecting any set of f~, it is easy to test any hypothesis about the analytical form of any set of data using this method.

In our case, since our original hypothesis was based on cycloidal velocity profiles at the joints, we choseft as a cycloid and for the remaining f~ we tried sinusoids, low-exponent polynomials, and higher-order cycloids. A comparison of the experimental data, pure cycloids

0,5 F'6y (Mg) 4

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0.5 FG* ~g) 1

TH(Mg-m) j

at.; TK (Mg-m) ~ 05

I ~ I I / - " i 0,0 0.t 012 0,5 0.4 0.5 0,6 0,7

Time (sec) Fig. 10. Forces and torques for fitted experimental data

and the best fitting curve for the knee and hip joint angles is shown in Fig. 9.

The expressions obtained for the fitted curves were then analytically differentiated to compute 0n, OK, 0"~, and 0"~ at each data point and were substituted in (31) through (34) of the Appendix to compute the ground forces and joint torques at landing. The results are shown in Fig. 10. The dotted line shows results obtained using conventional smoothing techniques and numerical differentiation of displacement data.

An additional advantage of using this curve- fitting method is that once the dominant functions are identified in the experimental data (using the relative magnitudes of their coefficients) their mathematical properties can be used to identify motion objectives. For example, it can be shown that (Guggenheim 1977) all cycloids are brachistochrones or minimum-time curves.

Furthermore, in trying to remove the effect of noise from the experimental data one does not have to guess the cutoff frequency as is done is most other smoothing techniques. In this method, since the relative dominance of all terms is explicitly apparent, one can judiciously drop higher-order sinusoidal terms with small coefficients as noise terms while knowing exactly how much error is introduced.

7 Concluding Remarks

The model of movement learning and control proposed in this paper offers a plausible explanation for various aspects of human limb movements. First, the model contains a simple type of preprogramming based on preplanned trajectories (cycloidal velocity profiles) defined by a small number of parameters. The inverse kinematics of the leg motion is computed only once and thus requires only minimum computation

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time, while also accounting for intrasubject variability. Second, the model supports a hierarchical structure and learning procedure allowing efficient combination of open and c losed l o o p con t ro l logics. Thi rd , the p r e p r o g r a m m i n g and curve f i t t ing p rocedu re p ro - posed here p rov ides a real is t ic s ta r t ing po in t for the s tudy of m o v e m e n t object ives t h r o u g h the m a t h e m a t - ical p rope r t i e s of specific ana ly t i ca l functions.

The results p resen ted here raise however a n u m b e r of ques t ions : Are mul t i a r t i cu la t e l imb movemen t s p l a n n e d in terms o f e n d p o i n t coo rd ina t e s o r in terms of j o i n t coo rd ina t e s? H o w does the inverse k inemat i c t r a n s f o r m a t i o n dea l wi th k inemat i c r e d u n d a n c y ? W h a t is the numer ica l c o m p u t i n g abi l i ty of the ne rvous sys tem? Keep ing these ques t ions in mind, it wou ld be in teres t ing to examine whe ther in t r ins ic ( jo in t coord i - nates) or extr insic (endpoin t coord ina tes ) reference systems c o r r e s p o n d to different m o v e m e n t objectives.

Appendix A) Nomenclature

m~ = mass of hip plus upper portion of body (kg) m2 = mass of thigh (kg) ma=mass of shank (kg) 12 = length of thigh measured from hip joint to knee joint

(m) T 2 = distance of center of mass of thigh from hip joint (m) 13 = length of shank measured from knee joint to ankle (m) T 3 = distance of center of mass of shank from knee joint (m) 14 = length of foot measured from ankle joint to tip of foot

(m) xn, yH=position coordinates of hip joint in a fixed flame of

reference (m) 2n, ~gn = velocity components of hip joint in fixed frame of

reference (m/s) 2~, Yn = acceleration components of hip joint in fixed frame Of

reference (m/s 2) xr, YF =position coordinates of foot in fixed frame of reference

(m) 2F, ))~ = velocity components of foot in fixed frame of reference

(m/s) s J~v =acceleration components of foot in fixed frame of

reference (m/s 2) OH, OK = displacement of hip and knee angles, respectively (rad) OH, 0r = time rate of change of hip and knee angles, respectively

(rad/s) UH, ~K = time rate of change of hip and knee angular velocities,

respectively (rad/s 2) g = acceleration due to gravity (9.8 m/s 2)

F~, F r =components of force acting on hip joint measured in fixed inertial frame (N)

G~.,Fvy = components of force exerted by the ground on the foot during landing (N)

TH, Tr= torques acting at hip and knee joints, respectively (N-m).

B) Kinematics and Dynamics

a) Kinematics. It is clear from the model chosen that once the motion of the hip is specified, for each value of the hip joint angle, On, the knee joint angle, OK, and their time derivatives, the

position, velocity and acceleration of the foot (or to be more exact the extremity of the shank) are uniquely determined. The x and y position of the foot, its velocity and its acceleration are given by:

xv = xn + 12 sin 0 u - la sin O, (1)

YF = Y ~ - lz cos O H + 13 cos O, (2)

s = 2n + IzOttcOSOH--130 e o s O , (3)

YF =.gH + 120n sin On - 13 0 sin O, (4)

2F=2n--120~sinOn+lEOnCosOn+IS)2sinO--13OcosO, (5)

r (6)

where

O=On+O K.

b) Inverse Kinematics. Conversely, if the motion of the hip is specified, On, OK, OH, OK, On, and OK can all be uniquely determined for any position, velocity and acceleration of the foot. The joint angle variables and their time derivatives are given by:

OK = arccos {[/2 + I~- X 2 - - YZ]/21fla} , (7)

0H=~P + z - f H , (8)

where

X = x r - x n ; Y=YF-Yn; q~=arcsin(13sinOK/d);

Z = arcsin(dv sin t/d) ; flu = arctan(xn/yH);

d=(lZ+lZ_21213cosOr)l/2; dr = (x v2 _}_ flF)2 1/2.,

f = ~ / 2 - f in - fie; fly = arctan(yp/xe)

OH = [13X sin O - l 3 Y cos 0]/1213 sin 0K, (9)

0 r = [12 2 sin OH-- 12 gcos OH]/1213 sin 0r -- On, (10)

On= [/3(X + 120~sinOn-130 2 sin O) sin O

-13(Y-120~cosOn+13(")2cosO)cosO]/I2IasinOK, (11)

OK = [/2(X + I20~ sin On - 13 0 2 sin O) sin O n -

- /2( 17-120~ cos O n + I a 0 2 cos O) cos On]/l 2l a sin 0 r - O n

(12)

c) Dynamics. The full dynamic equations for the model are derived using the Lagrangian method. The constraints during landing are modeled using Lagrange multipliers. Lagrange's equations for an n-DOF system are given by:

dt\c~gliJ ~ql =Q'+Ci ' (13)

where qi (i = 1, ..., n) are the generalized coordinates Qi (i = t . . . . , n) are the generalized forces C~ (i = 1 .. . . . n) are the constraint forces L is the Lagrangian of the system.

The Lagrangian is given by:

L = T - V , (14)

where T is the kinetic energy of the system V is the potential energy of the system.

For the model chose, the generalized coordinate vector is given by:

q=[q l q2 q3 q4]T=[XH YH OH OK] T"

The generalized force vector is given by:

Q=[Q1 Q2 Q3 Q4]T=[F~ F, T n TK] r

396

and the constraint force vector is given by:

C=J(q)~,,

where k is a vector of length m to be determined and J is the Jacobian matrix that relates the Cartesian coordinates of mass centers to the generalized coordinates.

The additional unknowns )~i (/= 1 ..... m) introduced by the constraint forces are compensated by the introduction of m constraint equations, which are of the form:

2 i ~ Ji jbqj:O (i=1 ..... m). j=l

The kinetic and potential energies of the system are given by the sum of the energies of each link as follows:

3 3 T= Z T~, V= Z Vii. (15)

i=l i=1

The various energy components are given by:

7"1 = 1/2mtv~ (16)

Tz=l/2m2[v~+2T2OH(icHcosOn+~nsinOn)+FzO~] (17)

T 3 = 1/2m3 {v~ + 2120n(~n cos 0n + 3)n sin 0n)

- 2/3 ~ [xn cos 6} + Yn sin 6}] 2"2 q- 12Or: -212~0n~ cos0r q-~3 ~z} (18)

where 2 -2 .2 v n : x n + yn

111 = m 19Yn, (19)

V2 = mzg(Yn-- T2 cos On), (20)

V3 = m39(Yn- 12 cos On + ~ cos 6}). (21)

During the swing phase of stepping there are no constraints on the motion of the leg. This means that m = 0, which in turn implies that C~=0 (i=1 ..... n) in (13).

Substitution of (16) through (21) in (13)-(15) then yields the equations of motion for the swing phase:

F~, = m x n + Jl(On cos O n - 02 sin 0n) + J2((02 sin 6} - ~ cos 6}),

(22)

Fy = M y n + J l(On sin O n + 02 cos On) + J2( ~ 2 COS 6} ~- ~ sin 6}), (23)

Tn=IlOn+ Jl(s ~nsinOn)+ I2(62sinOr-(OcosOn)

+ Jig sin On, (24)

Tx = I3 ~ -- J2(5r 6} + J)n sin O ) - 12(0/~ sin 0 r + O n cos 0~)

+ J2g sin 61, (25)

where

M = m l +mz+m3, Jl=rnffz+m31z, Jz=m3T3,

I1 =(m2~ +m3l~), I2=m31z~, I3=m3~.

During the landing phase of stepping, however, constraints are introduced on the motion of the foot. Mathematically, these constraints are given by:

xF = 0, (26)

YF = 0. (27)

When (26) and (27) are substituted in (3) and (4) we get the equations of constrains in terms of the generalized coordinates of the system:

.~.I[~)XHq-(~OH(12COSOH--T3cos6})q-~OK(--T3eosO)]=O, (28)

"~2 [~YH + 60( l 2 sin On-- 7-3 sin 6}) + 60K( -- T3 sin 6})] = 0. (29)

With these two constraint equations, the constraint force vector C becomes:

C=[21 22 0 0] T. (30)

Substituting (30) together with (16) through (21) into (13) we get the equations &motion for the landing phase:

Fax=J3( (OcosO-O2s inO)+ J4(O~sinOn-OncosOn), (31)

FGy = J3(O sin O - ~2 cos O ) - J4(02 cos O n + Ou sin On) + Mg,

(32) Tn = I40tl + 15(~ z sin OK -- O cos OK) -- J4g sin OH, (33)

Tr = 160-- 15(05 sin OK + 0"n cos OK) + J4g sin O, (34)

where

J 3 = M T a - J 2 , J4=(ml + m 2 ) l z - m j 2 ,

14=[mll~+m2(12--~)2], /5 = J4T3,

16 = (m 1 q'- m2) ~33 '1- m3( 13 - - ~ ) 2 .

Data used for various parameters of the model were:

rnl=50kg, m2=7.5kg, m3 =3.0kg,

12=0.75m, 13=0.60m, /2=0.35m,

13 =0.25m, /4=0.15m, g=9.8m/s z .

R e f e r e n c e s

Adams J (1984)Learning of movement sequences. Psychol Bull 96:3-28

Andre-Thomas A, Autgaerden S (1966) Locomotion from pre- to postnatal life. Clinics in developmental medicine, vol. 24. The Lavenham Press Ltd.

Beuter A, Cowden J (1986) Effect of partial transverse tarsal deficiency on movement kinematics: a comparative study (submitted)

Beuter A, Flashner H, Arabyan A (1986) Phase plane modeling of leg motion. Biol Cybern 53:273-284

Guggenheim HW (1977) Differential geometry. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York

Jury EI (1958) Sampled data control systems. Wiley, New York Marshall RN, Wood GA, Jennings LS (1984) Performance

criteria in normal human locomotion. Annual meeting of the Society of Biomechanics, Ann Arbor, MI

Nelson WL (1983) Physical principles for economies of skilled movements. Biol Cybern 46:135-147

Piaget J (1952) The origins of intelligence in the child. Routledge & Kegan Paul, London

Raibert MH (1978) A model for sensoirmotor control and learning. Biol Cybern 29:29-36

Thelen E, Bradshaw G, Ward JA (1981) Behav Neural Biol 32:45-53

Zajac FE (1984) Are natural movements optimum with special reference to posture, locomotion, and jumping? Neuroscienee 1984 Satellite Symposia: Organization of Movement, Society for Neuroscience

Received: August 19, 1986

Dr. H. Flashner Department of Mechanical Engineering OHE 430 University of Southern California Los Angeles, CA 90089 USA

Modeling of control and learning in a stepping motion

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