AN OPTIMAL CONTROL MODEL FOR MAXIMUM-HEIGHT HUMAN...

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AN OPTIMAL CONTROL MODEL FOR MAXIMUM-HEIGHT HUMAN JUMPING

MARCUS G. PAruDY*t, FELIX E. ZAJAC*, EUNSUP SIMS and WILLIAM S. LEVINEI

*Mechanical Engineering Department. Design Division, Stanford University. Stanford- CA 94305-4201. U.S.A.; *Rehabilitation Research and Development Center (la), Veterans Alfain Medical Center, Palo Alto, CA 94304-1200. U.S.A. and SElectrical Engineering Department, University of Maryland, College

Park, MD 20742, U.S.A.

Abstract-To understand how intermuscular control, inertial interactions among body segments, and musculotendon dynamics coordinate human movement, we have chosen to study maximum-height jumping. Rccausc thii activity presents a relatively unambiguous performance criterion, it fits well into the framework of optimal control theory. The human body is modeled as a four-segment. planar, articulated linkage, with adjacent links joined together by frictionkss revolutes. Driving the skeletal system arc eight musculotendon actuators, each muscle modeled as a three-clement, lumped-parameter entity, in series with tendon. Tendon is assumed to be elastic, and its properties an defined by a St-train curve. The mechanical behavior of muscle is descrihcd by a Hill-type contractile element, including both series and parallel elasticity. Driving the musculotendon model is a tint-order representation ofexcitation-contraction (activation) dm The optimal control problem ir to maxim&c the height reached by the center of mass of the body subject to body-segmental, musculotcndon. and activation dynamics, a xcro vertical ground reaction force at lift-off, aad constraints which limit the magnitude of the incoming neural control signalr to lie between xero (no excitation) and one (full excitation). A computational solution to this problem was found on the basis of a Mayna-Polak dynamic optimixation algorithm. Qualitative comparisons between the predictions of the model and previously reported experimental findings indiite that the model * reproduces the major features of a maximum-height squat jump (i.e. limb-segmental angular displacements, vertical and horizontal ground reaction forces, sequence of muscular activity, overall jump height. and final lift-off time).

INTRODUCDON

Motivated by the need to better understand how the central nervous system coordinates limb movement. Zajac and Levine (1979) have devoted much effort to using optimal control theory as a framework to study intermuscular control of multi-joint movement. They began by studying maximum-height jumping in cats (Zomlcfer et 01.. 1977; Zajac, 1985), and later pro- gressed to the same activity in humans (Levine et ol.,

1983a. 1987). Through a variety of increasingly complex models, they have gained insight into the theoretical and computational aspects of optimal control problems involving mammalian musculoskeletal systems.

In the case of a simple one-segment, planar baton, a complete analytical solution was derived (Levine et al.,

1983b). and a feedback optimal control was demon- strated. That is, the optimal control at any instant of time was expressed as a function of the state at that time. SpcciBcally, from certain regions of the state space, the optimal control involved applying maximum torque from the initial state until lift-o@. From many other states, however, the optimal solution was to first return the rod to zero angular displacement

Recebcd injinaiJbn 14 May 1990. tPracnt address: Dept. of Kincsiology and Health Educa-

tion, The Unfversity of Texas at Austin, Austin, TX 78712, U.S.A.

(the ground), and thcrcaftcr, to exert maximum torque until lift-elf.

With respect to more complex models of human jumping, a specific computational difficulty relates to the initial phase of propulsion where the entire foot remains fixed to the ground. Prior to heel lift-off, with the foot constrained from moving downward, the ground represents a dynamical discontinuity in the state space. For example, if the body-segmental model should have four degrees of freedom subsequent to heel lift-off, it would have only three while the foot remains flat on the floor. Such discontinuities violate the smoothness requirements of optimal control theory, and, consequently, earlier models (Levine et al., 1987) have limited themselves to the final propulsion (or bang-bang) phase of jumping.

By synthesizing information derived from expcri- mental measurements (limb-segmental motions, ground reaction forces, and clcctromyographic (EMG) data), several investigators have attempted to identify factors affecting limb movement coordination during jumping Grcgoirc ef ul. (1984), Bobbert et al. (1986a.b). van lngen Schenau et al. (1987). and Bob bert and van Inpn Schenau (1988) have all focused attention on the vertical jump in the hope of elucida- ting how muscles coordinate skeletal movement. By addressing issues of specific importance to jumping, their results have identified some of the major features characterizing this activity. For example, all joint angular velocities arc reported to decrease prior to lift-

1186 M. G. PANDY et al.

off (Bobbert and van Ingcn Schenau, 1988). and the sequence of lower-extremity muscular activation is shown to be proximal-to-distal (i.e. in the order hip, knee, and ankle) (Gregoire et 41.. 1984). Subsequent analyses of these data have also led to suggestions that overall jumping performance is heavily dependent upon biarticular muscle action (van Ingen Schenau et al., 1987).

A major goal of our ongoing research is to understand how intermuscular control, inertial interactions among body segments, and musculotendon dynamics coordinate a complex human motion. With this in mind, we have constructed an optimal control model for studying maximum-height human jumping which includes a reasonably detailed representation of both muscle and tendon. In addition, both single- and double-joint actuators are included in our analysis, as is the propulsion phase prior to heel lift-off.

Given that maximum-height jumping presents a relatively unambiguous performance criterion, it fits well into the framework of optimal control theory. Moreover, it is an activity characterized by bilateral symmetry which leads to a relatively simple representation of the body-segmental dynamical system. Most importantly, however, our motivation for using optimal control is founded upon the belief that it is currently the most sophisticated methodology avail- able for solving human movement synthesis problems (Chow and Jacobson, 1971; Ghosh and Boykin. 1976; Hatze. 1976). Optimal control theory requires not only that the system dynamics be formulated, but that the pcrformancc criterion bc spccificd as well. Thus, differences between model and experiment indicate deficiencies in the modeling of either the system dy namics or the performance criterion. The formulation presented in this paper allows us to simultaneously synthesize the time histories of all body-segmental motions, muscie forces, muscle activations, and incoming neural control signals.

THE MUSCULOSKELETAL MODEL

We modeled the human body as a four-segment, planar, articulated linkage, with adjacent links joined together by frictionless revolutes. A total of eight lower-extremity musculotendon units provide the actuation (Fig. I). An important facet of the computation of the optimal solution is how the dynamical constraint introduced by the foot resting flat during the initial phase of propulsion is treated. To circum- vent the computational difficulties that arise from such a constraint, we have placed a highly damped, stiff, nonlinear, torsional spring at the toes. The idea of modeling environmental contact with structures containing spring-damper combinations has been pro- posed by McGhee et al. (1979). However, this ap- proach introduces a pseudo-stiffness into the quations of motion which necessitates a decrease in the integration step size (Hatze and Venter, 1981). Never-

Fig. 1. Schematic reprcscntation of the musculoskeletal model lor the vertical jump. Symbols appearing in the diagram arc: solcus (SOL). gastrocncmius (GAS), other plantarflcxors (OPF). tibialis anterior (TA), vasti (VAS), rcctus fcmoris (RF), hamstrings (HAMS), and glutcus maxi-

mus (GMAX).

theless, with an appropriate stiffness and damping constant (see Appendix 1). the torsional spring serves its intended purpose by effectively modeling the foot-floor interaction prior to heel-off.

Body-segmental dynamics

The dynamical equations of motion for the four- segment model (Fig. 2) were derived using Newton’s laws. In vector-matrix notation, these may be expressed as:

A(e)ibB(e)b2+C(e)+DM(e)P’+T(e.8) (I)

where 9. b, a are vectors of limb angular displacement, velocity, and acceleration (all are 4 x I); T(0, & is a (4 x 1) vector of externally applied joint torques (for now it contains only the moment applied to the foot segment from the damped torsional spring); P’ is an (8 x 1) vector of musculotendon actuator forces; M(B) is

a (3 x 8) moment-arm matrix formed by computing the perpendicular distance between each musculotendon actuator and the joint it spans; A(8) is the (4 x 4) system mass matrix; c(e) is a (4 x I) vector containing only gravitational terms; s(e)@ is a (4 x 1) vector

describing both Coriolis and centrifugal etfects, where 82 represents df for i= 1, 4; and D is a (4 x 3) matrix which transforms joint torques into segmental

An optimal control model for jumping 1187

Fig. 2. Schematic rcpresenlation of the four-segment model for theverticaljump.m,.m,,m,.m,are thelumpcd massesof Ihe foot. shank, thigh. and HAT (head, arms. and trunk) respectively; I,. I,, I,. I, are the mass moments of inertia of the foot, shank. thigh, and HAT respeCtively. Body-scg-

mental parameters arc specified in Appendix I.

torques. The details of Appendix I.

Muscuhtmdon Bynumics

equation (I) arc given in

Each musculotendon actuator was modeled as a three-element, lumped-parameter entity (muscle), in series with tendon (Fig. 3). The mechanical behavior of muscle was described by a Hill-type contractile element which models its force-length-velocity character&&. a series-elasticelement which models its short- range stitmess, and a parallel-elastic element which models its passive properties. Tendon was assumed to be elastic, and its properties were represented by a stress-strain (U-C) curve. Other assumptions implicit to the musculotendon model were that all sarcomeres in a given fiber are homogeneous, all muscle fibers reside in parallel and insert at the same pennation angle on tendon, and muscle volume and cross-section remain constant. For a review of musculotendon dynamics, properties, and modeling, see Zajac (1989). Under these assumptions, Zajac et al. (1983) derived a first-order differential equation relating the time rate of change of tendon force to musculotendon length and velocity (IHT, uMT), muscle activation [a(t)], and tendon force (PT):

~=~~PT.I~T,L.MT,.(t),: O<o(r)<l. (2)

The details of equation (2) are given in Appendix 2.

Fig. 3. Schematic representation of the musculotendon model. Note that: IVT = I’+ I” cos 1: I” = 15” +F”; 10” sin z, = IM sin z = M’= const.; P’= P’ cos 1: where IwT is the length of the musculotendon actuator. I” and Ir are the lengths of muscle and tendon respectively: tSE and F” are the lengths of the series-elastic and contractile elements; P” and P’ are muscle and tendon forces; z is the pennation angle of muscle; LzT is tendon stilTness: kSE and k’” are the stilTness of the scries- elastic (SE) and parallel-elastic (PE) elements: CE and MT dcnotc the contractile element and musculotcndon ac1uator rcspectivcly: W (;I constant) represents muscle thickness; It. z,, are the fiber Icngth and pennation angle at which peak isometric force is dcvelopcd: and a(r) dcsignatcs activation of

the contractile elcmcnt.

Excitation-~ontruction dynumics

To describe the time lapse between the incoming

neural signal [mu& excitation, u(t)] and muscle activation [a(f)], WC have constructed a lirst-order equation:

4l)=(ll~,i..)(l -“)u(t)+(llT,.II)(u,i”-u)[l -u(O];

lJ(l)=O, I. (3)

Here. u(t) is assumed to be the net neural control signal to the muscle [i.e. we do not dissociate the ‘net’ firing rate control of a muscle from the recruitment control (Zajac. 1989)]. Also, T,~= and 7r.ll are rise and decay time constants for muscle activation respectively, and a,,,,” is a designated lower bound on muscle activation. introduced to cope with problems associated with inverting the force-velocity curve of muscle at low activation levels (Levine et al.. 1989; He, 1988). Note that the model is only used with u(f)=0 or I (see Appendix 3). Equation (3) may not accurately model activation at intermediate values of u(r). Appendix 2 gives a more detailed explanation of equation (3).

Muscubtcndon properties and muscu/o.~krhd

yfwmelry

Parameters defining muscle properties (i.e. peak isometric force and the corresponding pennation angle and length of the muscle fiber) for each of the eight musculotendon units were estimated from data reported by Wickiewia et al. (1983) and Brand et al. (1986). The linear U-E curve for tendon was specified using values of elastic moduli obtained from Alex- ander and Vernon (1975). Woo et al. (1982). and Butler

I188 hi. G. PANDY et al.

et al. (1984). while cross-sectional areas were estimated from anatomy textbooks if tendon had a well defined component external to the muscle. Otherwise, tendon cross-sectional area was chosen to give a reasonable strain at peak isometric force [i.e. in the range Z-6% which is well below the allowable limit (10%) defining tendon rupture (Zajac, 1989)]. Table 1 presents muscle and tendon parameters used by our model.

The musculoskeletal geometry of the model (musculotendon origin and insertion sites) was defined on the basis of data reported by Brand et al. (1982).

Table I. Muscle and tendon properties used in the model. All symbols are defined in Appendix 2

Muscle Tendon

Actuator

SOL OPF TA GAS VAS RF HAMS GMAX

(i:g) /:, PNS 1: s: (m) (%)

20.0 0.034 4235 0.360 2.5 10.0 0.036 3590 0.405 2.6 5.0 0.070 1400 0.265 2.7

12.0 0.062 2370 0.411 3.9 10.0 0.090 5400 0.206 3.0 14.0 0.075 930 0.323 2.6 9.0 0.106 2350 0.390 2.6 0.0 0.182 2650 0.090 5.3

normalized isometric

Rather than give details of the effective origin and insertion sites, we instead show plots of maximum isometric moment vs joint angle for the ankle. knee, and hip (Fig. 4). Also given in Fig. 4 are the corresponding experimental moment-angle data reported in the literature (e.g. Smidt, 1973).

The agreement between model and experiment (compare heavy solid lines with individual data points in Fig. 4) is reasonable for the ankle and knee, with the experimental data [eg. Inman et a/. (198 I) in Fig. 4(a)] in some cases being offset by as much as 20” from the moment generated by the model. Such differences may be due to experimental error (e.g. errors in joint angle measurement). However, larger differences between model and experiment are apparent at the hip, particularly as full extension is approached [compare heavy solid line with data points in the region 160-180” in Fig. 4(c)]. Given that large errors are associated with measurements of both moment and joint angle at the hip (in comparison with those at either the knee or ankle), for now we have elected to retain the muscle and tendon properties given in Table I (rather than adjust these to obtain a better match between model and experiment-see Results).

TOTAL

join1 angle (deg)

Fig. 4. Normalized isometric moment-angle curves of the musculotendon model. Heavy solid line is the sum ofall the extensor moments at the ankle (a); knee (b); and hip(c). Note that theshaded bars represent the range of angles covered during a maximum-height squat jump. Also. note that hip angle= 180+0,-O,: kneeangle- 180-0, +O,; and ankleangle= 180-0, +O,. where0,. 0,. O,.O, aresegmental anglesdefined in Fig. 2. In each case, muscle is assumed to be fully activated, and the moments generated are those only due to active muscle. The curves given for double-joint muscles were obtained by varying the angle at one joint, while holding that at the other joint constant. These constant joint angles were 90’ at the ankle. and 180” at both the knee and hip. For all muscles, tendon slack lengths were adjusted until realistic moment-angle curves were obtained. Each curve has been normalized by the peak moment at each joint generated by all the flexor and extensor muscles. These values were 220. 173. and 228 Nm for the ankle, knee, and hip respectively. Similarly, experimental data were normalized by their respective peak values of moment. The experimental data were obtained from: (a) ankle: 0 Inman er al. (1981); 0 Sale et al. (1982); (b) knee: 0 Smidt (1973); 0 Lindahl et 01. (1969); (c) hip: x Nemeth et ol. (1983)(malesh 0 Nemeth et al. (1983) (females);

0 Waters ef al. (1974).


MusculotrndinoskeIetaI dynamics

The dynamical quations for the overall musculo- tcndinoskelctal system arc therefore:

i9-A(6)‘‘[B(6)V+C(8)+DM(B)Pf+ Z-(8,8)] (4)

fi:=A(@, b, P:, aJ i= 1.8 (5)

W==(W~)(l -ar)ur+(llrc.u)(a,i.-a,)(l -uJ

i--1,8. (6)

vertical ground reaction force). We imposed intermediate constraints on the limb-segmental angles in order lo prevent joint hyperextension. These, however, were found to be inactive. The constraints which remain active are:

and

Obu(t)< 1 (8)

The state vector, defined by [x,,x,.x,.x,J’ = [g, 8, PT, a]’ (the prime ( ’ ) indicates transpose of a vector). is composed of 24 elements: four angular displacements tI(. i= 1, 4, four angular velocities 0,. i= I, 4. eight muscle activations a,, i= 1.8. and eight musculotendon actuator forces P:, i= 1.8. Each control of the input control vector u=[u,, u2, . . . , u8 J’ is coupled lo muscle activation through excitation- contraction dynamics [equation (6) 3, but otherwise is decoupled from the musculotendon dynamics (Fig. 5). Muscle activation is, on the other hand, coupled to musculotendon dynamics [equation (S)] (Fig. 5). Finally, muscle force is the interface between musculotendon dynamics and body-segmental (skeletal) dynamics [equation (4)] (Fig. 5).

F,(e. 8,3)&= t mi(‘i;,+g) =0 (9) 151 II

where m, is the mass of the ith segment, i;c, is the vertical acceleration of the center of mass of the ith segment, F,(& 8, a) is the magnitude of the vertical ground reaction force, and I,, indicates that each quantity is evaluated at the final lift-off time.

Thus, the optimal control problem is to maximize equation (7). subject to the given initial conditions x(O)= x0 and equations (s)--(6), with equations (8) and (9) acting as interior and terminal path constraints, respectively.

METHODS

Oprimiration oj the initial states

OPTIMAL CONTROL PROBLEM

For maximum-height jumping, we chose the height reached by the center of mass of the body to be the measure of performance. Previously, Levine el 41.

(1983b) have shown that there is little difference between this objective function and one which maxim- izes the height reached by the uppermost point of a one-segment model. They have also verified that such a result is independent of the number of segments representing the skeletal system. Therefore, our performance criterion is defined as:

At time t =0 the body is assumed lo be static. in a prespecified squat position. Thus, muscles develop moments at the ankle, knee, and hip to counteract the effect of gravity. However, with eight muscle groups, there are an infinite number of combinations of actuator forces that will generate these three moments. Therefore, we are presented with the classical muscle-force, joint-moment redundancy problem which is usually handled by staticoptimization techniques (Crowninshield, 1978).

Ju4 4 t,)= U’, )+ WQYZS (7)

where Y,(t,) and f&,) are the position and velocity of the center of mass of the body at time t,, the instant at which lift-off occurs, and g is the gravitational acceleration constant.

During the initial squatting position. we hypo- thesize that our subjects distribute their muscle forces such that the sum of the squares of muscle stress is minimized. The quadratic objective function is thus:

J= i V’:lPd2 111

(W

The constraints, which define the problem, are where Pf is the value of the ith musculotendon force, the dynamical equations [equations (4)-(6) 1, a set of and P,,, is the peak isometric strength of the ith muscle. inequality constraints which bound the magnitude of Note that the parameter P,, is directly proportional each neural control signal, and a terminal equality to muscle cross-sectional area. Therefore, the static constraint that specifies the instant of lift-ofT(i.e. a zero optimization problem was to minimize equation (10)

u(t) Excitat$-CbC~;action a(t) Musculotendon ph _ Skeletal e(t). b(1). m

- Dynamics Dynamics i I

ho

Fig. S. Block diagram showing interactions among the major compartments of the musculotcndino- skeletal model. Note that excitation-contraction dynamics is dccoupicd from musculotcndon and skeletal

dynamics.

II90 M. G. PANDY et al.

subject to three linear equality constraints (i.e. moments exerted at the ankle. knee, and hip), together with eight inequality constraints which preclude muscle forces from becoming negative:

O<Pf<x. (11)

A solution was obtained using a quadratic programming algorithm based upon the active-set, null-space method (Gill et al., 1984). Having found a set of optimum muscle forces, equation(2) was then used (with P’=O under static conditions) to iteratively solve for the corresponding muscle activation levels. We found the initial value of P: to be always lower than the peak isometric strength PO, of muscle. We note here that we are currently attempting to solve for the initial conditions which maximize jump height. Our preliminary results indicate that the optimal control solution is more sensitive to changes in the controls u(t) than to changes in the initial muscle forces.

Computation o/the opf imol controls

The optimal control problem, as formulated here, is ‘bang-bang’ (i.e. the optimal controls can only take values of zero or one). An important feature of the dynamical equations [equations (4)-(6)] is that the time rate of change of the state vector x is linear in the controls u. which is a consequence of our lirst-order model for excitation-contraction dynamics [i.e. the time derivative of muscle activation is linear in the controls in equation (3)]. As a result. the system Hamiltonian is linearly dependent upon the controls and the optimal controls must be bang-bang (see Appendix 3). However, singular controls (i.e. controls between 0 and I, see Appendix 3) are theoretically possible if it is assumed that the foot is both rigid and stationary on the ground, as may occur at the start of the jump. By modeling this phase, when multiple contact occurs between the foot and the ground, using a damped torsional spring, we force the solution to be bang-bang. Nevertheless, singular controls, if they should occur, would be detected because the controls in our solution would switch frequently (compared to system dynamical response time) between 0 and 1. Only rarely have we found the controls (i.e. the neural excitation signals) to exhibit such evidence for singular controls.

To solve the optimal control problem, we imple- mented a modified version of an algorithm developed by Polak and Mayne (1975) (see Sim (1988) for details). At each step in the iteration, the computation was begun with a forward integration of the state quations using an arbitrary initial guess for the controls. The forward integration proceeded until the terminal equality constraint was met (i.e. the vertical ground reaction force becomes zero), at which point the adjoint dynamical equations were integrated backwards in time using a boundary condition on the co-states computed from the state at lift-off. The co-

states pertaining to muscle activation levels were then used to find a new set of controls which increased performance [see Sim (1988)].

The vector of co-states is of considerable importance because it can be used to assess the optimality of our solution. Since the optimal controls are bang-bang, we need only to solve for the switch times which maximize jump height. It is an important fact that the optimal switching times are directly related to the sign of the co-states of muscle activation. and maximizing the system Hamiltonian amounts to choosing the controls on the basis of the sign of these co-states. Specifically. if a co-state has positive sign, then the optimal control must be u = I to maximize the Hamiltonian; similarly, a negative value of the co-state means that the optimal control must be u=O. Clearly then, at the optimum, the controls must switch at precisely the instants at which the co-states for muscle activation change sign. We used this fact to assess the accuracy with which our algorithm is able to locate a local maximum.

RESULTS

Limb-segmental motions, ground reactions, and optimal

controls

Qualitative comparisons of model predictions with published experimental results (e.g. Grcgoire et ul.. 1984; Bobbert and van lngen Schenau. 1988) indicate that the model can reproduce the major features of a maximum-height squat jump (i.e. limb-segmental angular displacements, vertical and horizontal ground reactions, sequence of muscular activity, overall jump height, and final lift-08 time).

The simulated jump height (net vertical displacement of the center of mass of the body from standing) and lift-00 time were 33 cm and 0.5 s respectively. These compare well with experimental values re- corded during vertical squat jumps performed by male volleyball players. For example, Komi and Bosco (1978) have calculated average jump height to be 37.2 + 3.7 cm, while lift-off times are typically in the range of 0.4-0.5 s.

Figure 6 presents the limb-segmental angular displacements and velocities for the foot, shank, thigh, and trunk. In agreement with published experimental results (e.g. Komi and Bosco. 1978; Bobbert and van lngen Schenau, 1988). the model undergoes countermovement, achieved primarily by a downward motion of the shank and thigh [Fig. 6(b) and(c)]. The ground reactions generated by the model (Fig. 7) are also qualitatively consistent with published experimental results. Peak magnitudes of the vertical ground reaction lie in the vicinity of 2.5 times body weight, whereas the horizontal ground reaction (i.e. the fore-aft component) is less than 30% of body weight (Komi and Bosco, 1978). Evidence for the existence of a preparatory countermovement is also visible in the vertical ground reaction, which shows a decrease in force during the first 40% of the jump (i.e. in Fig. 7, at


force (8 body Jo0

I

weight)

hori:ontal

wt , , , , I.12 0 20 40 60 do IW

% o/ground contact rime

Fig. 6. Limb-segmental angular displacements (solid lines) and velocities (dotted lines) generated by the optimal control model. Note that the foot. shank, thigh, and HAT angles correspond with 0,. 0,. 0,. 0, shown in Fig. 2. and that 100 % of ground contact time coincides with lift-off. Notice that each segmental angular speed (the magnitude of xg-

mental angular velocity) increases just prior to lift-off.

first, the vertical reaction fails below body weight, due to a downward acceleration of the center of mass of the body). A more detailed comparison of the response of the model and experimental results obtained for several subjects performing a maximum-height squat jump is given by Pandy and Zajac (1991).

We remark that the high frequency ripple evident during the initial phase of propulsion (Fig. 7) is an artifact of ,the spring which is used to facilitate computation of the optimal controls while the foot is on the ground. Essentially, the action of the lower-extremity musculature is to try to force the heel into the

I 0 20 do c.2 do ;.v

f/o of ground contact time

Fig. 7. Vertical (heavy solid line) and horizontal (light solid line) ground reaction forces generated by the model. Because the vertical force decreases at first. the model suggests that a countermovement is necessary. even in a squat jump. to jump as high as possible. Note that 100 % of ground contact time

coincides with lift-off.

ground, while the spring exerts an equal and opposite torque on the foot. As the spring is made stiffer, the frequency of the ripple increases and its amplitude decreases, but the trade-oft’ is that computation time increases.

The fact that the optimal control solution shows a proximal-distal sequence of muscle activation further increases our confidence in the response of the model. Figure 8 shows the optimal controls. together with the time histories of muscle forces predicted for four of the musculotcndon actuators (i.e. SOL, GAS, VAS, and CMAX). It is clear that muscle activation is sequenced in the order hip, knee. and ankle. We found that optimal performance (i.e. maximum jump height) dc- mands that the hip extensors GMAX and HAMS be activated first. followed by the knee extensors VAS and RF, then by the ankle plantartlexon SOL, OPF, and lastly by GAS (compare the heavy solid lines in Fig. 8). This trend is very consistent with experimental EMG results reported for the vertical jump (Gregoire et a!., 1984; Bobbert and van lngcn Schcnau, 1988), where it is shown that lower-extremity muscles are not activated simultaneously, but rather in the above noted sequence.

While the model successfully reproduces the major features of a maximum-height squat jump, shortcom- ings are evident. Even though the stick figures in Fig. 6 indicate that the body-segmental motions are coordinated (i.e. no joint hyperextension), discrepancies exist at the beginning and end of the jump. In Fig. 6(d), trunk countermovement in the model prior to upward propulsion is hardly noticeable (i.e. less than 5”). whereas experimental studies report downward trunk rotations of up to 25” (Pandy et al., 1988). WC hypoth- esize that this anomaly is due to the inability of the model to exert a sufficiently large moment at the hip (see Pandy and Zajac (1991) for a more detailed discussion).

Another contradictory feature of the response of the model is the increasing limb-segmental angular speeds

1192 M. G. PANDY et al.

muscle force tm~b (N)

neural input I

n looo- CMAX I/ \

I

0

96 of prorcnd conract rime

Fig. 8. Time history of neural excitation input signals for four musculotendon actuators (heavy solid lines) and their forces (light solid lines). Note that 100 % of ground contact timecoincides with lift-oliand 0 % (vertical dashed line) with the time the jump starts. Prior to 0 % time. muscle forces arc constant to maintain the body statically in the squat. Notia

the proximal-distal scqucnce of excitation of muscles.

just prior to lift-off [see dotted lines in Fig. 6(a)-(d) during the final 10% of ground contact time, and compare with Bobbcrt et al. (1986a) and Bobbert and van Ingen Schenau (1988)]. In fact, it has been hypo- thesized that limb-segmental angular speeds decrease during human jumping because of flexor muscle activity (Bobbcrt and van Ingen Schenau, 1988). Our results indicate that single-joint flexor muscles in the lower extremity should be inactive immediately prior to and at lift-off, and that all extensor muscles be fully activated until lift-OR Since this result is robust, we believe that it is fundamental to the structure of our optimal control problem. [Though Fig. 8 shows SOL to be de-excited prior to lift-off, we do not view this as a contradictory result since the model drives SOL into a region of the force-length curve where zero force is generated (i.e. at ankle angles greater than ISO”).]

One important finding was the high sensitivity of the optimal control solution to changes in the neural excitation of VAS. Figure 9 presents (in stick-figure form) the limb-segmental angles generated by our model under the condition that the optimal switching

time for VAS [see Fig. I(VAS)] is delayed by just ten per cent. These results indicate that even small changes in the timing of VAS are sufficient to alter the motion

Fig. 9. _.

Stick figures showing how uncoordinated the body’s mouon becomes when VAS excitation is delayed by just IO % from the optimal. Note that 100% of ground contact time

coincides with lift-off.

of individual limb segments to the point that coordination is significantly reduced. In particular, note that both foot and trunk angular displacement exceed 90” (i.e. pass through the vertical), while the thigh undergoes an abnormal decrease in velocity during upward propulsion (not shown in Fig. 9). By contrast, coordination seems to be much less sensitive to changes in the control of other muscles. In the case of GAS, for example, exciting it 10% earlier or later than optimal has little c&t on overall response. In this case, at lift-off, all joint angles are within 3” of the optimal motion, and performance decreases by less than 5%.

An important component of our optimal control solution is the vector of co-states, where each element of the co-state is associated with one of the 24 states (Appendix 3). The value of the co-state at any instant of time represents how optimal jump height would change given a small instantaneous (positive) change in Phe associated state. That is, if the state could be instantaneously changed at time 1,. to< Cl <t,, we would have a new optimal control and trajectory from t, to a new lift-off time (t,). The height of the new optimal jump would also be different. either higher or lower. To first order, the new performance would be

.I “.W =JJo,d+A+‘(t,)Ax(tt) (12)

where Ax(t,) is the instantaneous change in the state x(t,) at C= I,. and the other symbols are defined in Appendix 3. Given the dynamical model of the mus- culotendinoskeletal system used in this study, however, instantaneous changes in the states (i.e. muscle force, activation, and segmental angular position and velocity) are assumed to be physically unrealizable [see equations (4)-(6)]. Only the controls, the muscle excitations u(l). can change instantaneously.

In general, at any time during the jump, including lift-off. jump height is most sensitive to changes in angular displacement. A comparison of the results in Fig. 10 reveals that the co-state associated with thigh angle is at least an order of magnitude larger than the co-states associated with thigh angular velocity, vasti force and activation. Changes in limb-segmental angu-


normalized 1 costates a

% of ground contact time

Fig. 10. Time history of four of the twenty-four co-states, normalized by the magnitude of the corresponding state at each instant. The units in each case arc m %-I change in state. The value of each co-state at lift-off (100 % of ground contact time) shows how scnsitivc optimal jump height [equation (7)] would bc to a small incrcmcntal change in the associated state at lift-oft Notice that the co-state associated with thigh angle afTccts optimal jump height much more than

the other co-states.

lat displacements therefore have the greatest potential for improving jump height. This is especially noticc- able at lift-off, where a small positive change in thigh angular displacement, for example, will lead to a decrease in jump height [i.e. the co-state thigh angle has a negative value at 100% of ground contact time, Fig. 10(a)]. Since jump height depends only on the segmental angular displacements and velocities at lift- OR [quation (7)]. only the co-states associated with these states are non-zero at lift-otT (Fig. 10(a), (b) at I&/O ground contact time), whereas the co-states associated with vasti force and activaiion are zero at that time(Fig. IO(c).(d) at 100% ground contact time). Though optimal jump height is sensitive to the angular velocity of the segments at lift-off, angular velocity is less important than angular displacement [compare the magnitudes of the co-states associated with thigh angle and thigh angular velocity at 100% ground contact time, Fig. IO(a). (b)].

Finally, Fig. 11 is a plot of the time history of muscle activation co-states (light solid lines) for four musculo-

tendon actuators: SOL, GAS, VAS, and GMAX. Also shown are the corresponding control signals (heavy

muscle activation _;;!$, T

01

i I’

L I

I I I

.mI 0 10 40 60 so ,a,

46 of ground contact time

Fig II. Neural excitation input signals (heavy solid lines) and muscle activation co-states (light solid lines) pmdictcd by the model. The co-states arc not normalized. Note that, in each case. the curves intersect each other at acre. indicating that the solution is indeed optimal. One hundred per cent of

ground contact time coincides with lift-otT.

solid lines). Since the controls switch from zero to one at the same instants that the co-states for muscle activation change sign (note the intersection of the light and heavy solid lines in Fig. ll), the computed optimal controls are very close to the theoretical optimum. Note also that, for some muscles (e.g. SOL), the controls are character&d by narrow spikes in regions where the co-states indicate muscle inactivity (i.e. the co-state for activation of SOL is zero prior to lift-off in Fig. II, indicating that neural excitation should also be zero). It is for this reason that our solution is, strictly speaking- very near, but not exactly at, the theoretical optimum. Nevertheless. the accuracy of our solution is sutlicient to justify its use in more detailed analyses of jumping (Pandy and Zajac, 1991). For example, a necessary condition for an optimal control is that the Hamiltonian be zero at any instant throughout the jump. Our computation pro- duces values no larger than 0.04 over the entire jumping cycle (values in the order of OS-I.0 would indicate non-optimality).

DISCUSSION

Given that our model is nonlinear and of high dimension (i.e. it has a total of 24 states), and that its

1194 M. G. PANDY et al. _

response is particularly sensitive to changes in the input excitation signal for VAS (Fig. 9). it is not surprising that. for reasonably complex models of the human musculoskeletal system, heuristic solutions for body-segmental motions (i.e. guessing the input controls) often prove unsuccessful. In contrast, optimal control theory is an exceptionally valuable tool because it not only defines the time history of all body- segmental motions, muscle forces, and muscle activations, but it simultaneously delivers a fully coordin-

ated result (compare the stick figures in Fig. 6 with those in Fig. 9).

Due to the complexity of our musculotendino- skeletal model, the task of finding an optimization algorithm that converges is a particularly difficult one. Fortunately, Sim et al. (1989a) have been able to modify the Mayne-Polak algorithm to efficiently solve optimization problems that are monotonic (i.e. steadily increasing or decreasing) in the controls. We are applying this scheme to study other human motor tasks in addition to jumping (e.g. Sim et al. (1989b) have been studying the optimal control of bicycle pedaling). There are, however, some computational problems that arise because of our model for muscle. One particular problem relates to the difficulty involved with inverting the force-velocity curve of muscle (i.e. compute muscle velocity given muscle force) at low activation levels. Under these conditions, the routine is forced to decrease its integration step size until this computation becomes possible. At best, this results in a significant increase in the time taken to converge to a solution. At worst, the algorithm reaches a point where it gets ‘stuck’ (i.e. the required integration step size approaches the limits of machine pre- cision), in which case the computation is stopped.

Under certain circumstances, the problem of inverting the force-velocity curve for muscle also leads to inaccuracies in our computation of the optimal controls. During a countermovement jump (Pandy et al.. 1988). for example, the activation level of most muscles is initially low (i.e. the model begins from an upright position with most of its muscles relaxed). In this instance, we have discovered large errors in the co-state values computed during the backward integration. Since the values of the controls needed to maximize the system Hamiltonian are chosen directly from the

computed co-states for muscle activation, any inaccur- acy in the computation of these co-states leads directly to inaccuracies in the estimated optimal controls. ln fact, for the countermovement jump, we have found the computed switch times compare poorly with those instants at which the co-states for muscle activation change sign. Consequently. we have more conti- dence in our optimal control solution for squat jumps than previously reported results for the countermovement jump (Pandy et al.. 1988). However, even for the countermovement jump, the computed controls lead to a coordinated jump, which otherwise might be difficult to find.

Our results indicate that the response of the model

is at least qualitatively similar to experimental data reported for jumping (Gregoire et 01.. 1984; Bobbert and van Ingen Schenau, 1988). For example, the magnitude of the vertical reaction and the sequence of muscle activation generated during propulsion are both consistent with experimental trends reported for the vertical jump (e.g. Bobbert and van Ingen Schenau, 1988). However, to assess the response of the model more closely, we have collected experimental results for several subjects instructed to perform a maximum- height squat jump (see Pandy and Zajac, 1991). These data further support our contention that the model is now sufficiently accurate to justify a detailed analysis of the optimal control solution.

Our future work involves investigating the dependence of jumping performance on muscle speed, strength, and tendon stiffness. Examining the dependence of performance on muscle-fiber speed is relatively straightforward because it only requires a change in the maximum intrinsic shortening velocity of muscle (now assumed equal to IOs- ‘). Since this parameter regulates the area under the force-velocity curve for muscle, we must expect it to influence jump height considerably (i.e. increasing maximum shortening velocity results in greater levels of energy liberated by muscle). In fact, in studies relating to the sensitivity of performance to small changes in muscle and tendon properties, Sim (1988) found that jump height is most sensitive to small changes in the maximum shortening velocity of VAS. In general, specifying large changes (i.e. 3oo-400%) in the maximum shortening velocity of muscle will enable us to predict how the intrinsic ability of muscle to shorten (i.e. fast- vs slow-twitch fibers) affects jumping performance. Similarly, we can use our model to investigate the dependence of jump height on muscle strength. In this case, altering the peak isometric strength of muscle, in addition to individual body-segmental masses, will enable us to find how a person’s ‘strength-to-weight ratio’ affects performance. We can therefore use our model to investigate the differences between ‘strong’ and ‘weak individuals within the framework of achievable jump height.

Finally, a factor commonly thought to influence jumping performance is tendon stiffness (Bobbert et al.. 1986a; Komi and Bosco, 1978). Specifically, experimental studies have reported that jump height is very dependent upon the stiffness of tendinous structures in the human triceps surae (Bobbert et al., 1986a. b). Investigating the effect of ankle plantarticxor tendon stiffness amounts to varying either tendon slack length 1,‘. or tendon strain at peak isometric force cl. (It is important to remember that parameters such as tendon slack length cannot be changed indiscriminantly because they directly alter the shape of the moment- angle curves.) Thus, we will be able to find how large

changes in tendon stiffness for SOL, OPF, and GAS

affect performance. Given that tendons spanning both

the knee and hip are much shorter (and therefore

stiffer) than those crossing the ankle (Zajac, 1989). we

assume that overall jumping performance is much less muscular-skeletal system of a cat hindlimb. Ph.D. thesis,

sensitive lo tendon stitrncss of the more proximal Electrical Engineering Department. University of Mary-

muscles. By introducing such changes to our model, land.

we will be able to quantitatively assess which factors Ingen Schenau. G. J. van, Bobbert, M. F. and Roundal, R.

dominate performance (i.e. whether jump height is H. (1987) The unique action of biarticular muscles in complex movement; 1. Anal. 155, l-5.

more significantly influenced by intrinsic muscle-fiber Inman. V. T.. Ralston. H. J. and Todd. F. (1981) Human ’ speed, strength, or ankle-plantarflexor tendon stiff- W&ing. Williams and Wilkins, Baltimore.‘

ness). The ability to address such global issues makes Komi, P. V. and Eosco. C. (1978) Utilization of stored elastic

the use of optimal control theory especially appealing. energy in leg extensor muscles by men and women. Med. Sci. Sports 10(4), 261-265.

Acknowledgements-We thank Eric Topp for assistance with Levine, W. S.. Christodoulou. M. and Zajac. F. E. (1983b) On

computer simulations and David Delp for his help wifh propelling a rod to a maximum vertical or horizontal

figure preparation. This work was supported by NIH grant distance Auromorico 19, 321-324.

NS17662, the Alfred P. Sloan Foundation, INRIA, and the Levine, W. S., He, J., Loeb. J.. Rindos, A. and Weytjiens, J.

Rehabilitation R & D Service, Dept. of Veterans’ Affairs. (1989) Analysis of some standard assumptions in muscle modeling suggests some enhancements and muscle properties (in preparation).

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887-898. Rahmani. S. (1979) Dynamic modeling of human locomo-

Bobbert. M. F. and Ingcn Schenau. G. J. van (1988) Co- tion. Proc. 1979 Jr Aufom. Conlr. Co& pp. 405-413. ordination in vertical jumping. J. Biomcchunics 21. Denver, Colorado. 249-262. Nemeth. G.. Ekholm. J.. Arborelius, U. P.. Harms-Ringdahl.

Brand, R. A.. Crowninshicld, R. D.. Wittstock. C. E., Pedcr- K. and Schuldt. K. (1983) Influence of knee Rexion on

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Chow. C. K. and Jacobson, D. H. (1971) Studies of human Pandy, M. G.. Zajac. F. E., Hoy, M. G.. Topp. E. L.. locomotion via optimal programming. Murhl Siosci. IO, Tashman. S.. Stevenson, P. J. and Cady, C. (1988) Sub- 239-306. optimal control of a maximum-height, countermovement

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Gregoire. L.. Veeger. H. E., Huijing, P. A. and lngen Schenau. Sim. E. (1988) The application of optimal control theory for G. J. van (1984) Role of mono- and biarticular muscles in analysis of human jumping and pedaling. Ph.D. thesis, explosive movements. Inl. J. Sporls Med. 5, 301-305. University of Maryland.

Hatze, H. (1976) The complete optimization of a human Sim, E.. Levine. W. S. and Zajac. F. E. (1989a) Some results motion. MarhI Binsci. 2& 99-135. on modeling and computing the controls that produce

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I,, = distance of segment center of mass from diital end mr=massofsegtnenti 19, = angle that segment i makes with the horixoatal.

alI

A(B)= -a11cdI-4) a~,cos(4-~,)

-a,4cos(B, -e,)

0

B(e) = -a,*sin(g,-0,) a,,sit@,-e,)

-a,,sin(l,-8,)

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Zajac, F. E. (1985) Thigh muscle activity during maximum- height jumps by cats. J. Neurophysiology S3(4). 979-994.

Zajac, F. E. (1989) Muscle and tendon: propcrtics, models, scaling, and application to biomcchanics and motor control. In CRC Critical Review oj Biomedical Engineering (Edited by Boumc, J. R.), Vol. 17, pp. 359-41 I. CRC Press, Boca Raton.

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Zajac, F. E.. Topp. E. L. and Stevenson, P. 1. (1986b) A dimensionless musculotendon model. Proc. 8th A. Con/. IEEE &gag Med. Bio/. Sot.. pp. 601-604, Dallas-Ft Worth, TX. 7-10 Nov.

Zomlefer. M. R., Zajac, F. E. and Levine, W. S. (1977) Kinematics and muscular activity of cats during maximal height jumps. Brain Res. 126.563-566.

APPENDIX I

Body-segmental dynamics

The details describing the matrices defined in equation (I) are given below. Some definitions are:

I,=moment of inertia of segment i about its center of mass i,=length of segment i

-a,,w4-4) ~,,ww4) -a,.d4 -0.) al2 -a13=de,-W a,.we.-e,)

-a23W@,-f%) a33 -a,.wh -4) ~l.we~l) -~,~~~~(e,-e.) k 1

-a,* sin@, -e,) -a,,sin(B,-0,) -o,,sin(ll,-8.)

0 -a,,sin(8,-8,) a,,sin(g,-e.) -o,,sin(e,-0,) 0 -a,. sit@, - B.) -a*. sin(tJ, -ed) -o,,sin(tJ,-8,) 0 1

where

-c,gcose,

a@‘= cs c0se,

[ I -cacose, c&q cost&

2 0 0

D= 1 -2 -2 0

0 2 2 0 o-2 1

~,,=I,+m,If,+(m,+m,+m,)l~

~tl-14

01,-c, I 1

414-c* I I

411=1,+m,Ifi+(m,+m,)l:

aa3 -A

a14=c4 I I

a,,-I,+m,If,+m,l~

a,4-c4, 1

h4 - 14 + 4,

cl-m,I,,+(m,+m,+m,)l,

c2=md,+h+m&

c3=mJc,+m,G

c4=mJ,,.

The body-segmental parameters used for the skeletal model an as follows:

Segment m(ks) C,(m)

Foot 2.2 0.095 Shank 7.5 0.274 Thigh 15.1s 0.251 HAT 51.22 0.343

1, (m)

0.175 a435 0.400 0.343

Ukg ml)

:g 01126 6.814

The moment applied to the foot segment from the nonlinear, damped. torsional spring is defined as:

when e, c 0,;

otherwise F(6, 6)=0

An optimal control model for jumping 119-l

where:

k,= spring stiffness constant (9.0 x 10’ Nm rad-‘) c, = spring damping constant (2.0 x lo6 Nm rad -%‘) 8, = foot segment angle when foot is flat on the ground (34’).

Note that when 8, > 0,. 7(8.~=0, because when the heel is lifted OR the ground the spring is no longer in e&t. Note also that the term (k,-c,6:) is always positive when the foot is on the ground because the angular velocity of the foot scgmcnt 8, is very small at that time (i.e. 6, < 1 when 0, >9,).

APPENDIX 2

The musculotendon model

The dynamical quations describing the musculotendon and activation properties of‘an actuator are summarized below. Details can be found in 2!ajac et 01. (1986a.b). A treatise on modeling musck with a Hill-type model, and the development of a model similar to the one below is given in Zajac (1989). which cites the references relevant to these models (see also Winters and Stark, 1987). Before activation dynamics is summarized, a model ofmwculotendon contraction dynamics (see Fig. 3) is given. Some definitions are:

a(t)=muscle activation P’ = force in tth element I’- length of ith element ul= velocity of fth element k’=stiffncss of ith clement (I-F, SE, PE)

k”=k”+ksE==musck stiffness (a-0) kYT-(kT.kM/kT+kY)=actuator stiffness (a=O)

tT - tendon strain

where i-F. M, MT, PE. SE, or CE. cxccpt as noted.

The properties of each musculotendon actuator were derived from scaling generic muscle and tendon properties by five parameters. The five actuator-specific parameters are:

P, = peak isometric active muscle force If = muscle fiber length where P,, is developed a = muscle fiber pennation when P’ = I,” /~-tendon slack length eX = tendon strain when P’= P,.

Maximum intrinsic shortening velocity was assumed to bc

muscle independent (= 10~~‘). Thus. the maximum shortening velocity (V,) of a muscle was set qua1 to lOI,Ys-‘.

Passive muscle (the FE) was assumed to generate force P’L(I”) at lengths I’ > If. and activated muscle to generate an additional. active muscle force PC” (as given by the CE). Isometric active muscle force Pg[l”. o(r)] was assumed to be multialicative in the amount of muscle activation nltk i.e. Pgil”‘, u(t)] =P$tY).a(t), where PT,L(I”) is th; force-kngth curve of active muscle when dt)= I [i.e. when muscle k fully activated; Fig AI(b)]. The fo~vclocity relation of muscle is accounted for by assuming that the force FE generated by the CE is agected by its velocity rec. and is given by a pr(ucr) relation [or, quivakntly. by a velocity-force relation; see the ucr(~‘) curve in Fig Al(a)]. The force generated by the CE was assumed to be

F&=(P:,CII”. o(t)J/P& F”co’“)

=(P:.s(f”)..(r)/P,). prL(oCE) (A2.1)

which implies that the velocity of the CE at zero force. rcgardkss of muscle activation or length. is always equal to its maximum shortening velocity V,. The velocity of the CE was found by solving equation (A2.1) for 6’; i.e.

UC& -/! c~E/v%wo)l (A2.2)

whcref,( *) is the velocity-force curve in Fig. Al. Stiffness of active musck (the SE) was assumed to be

kS”=(loOP=+ loP,)/I;. (A2.3)

This stiffness implies that when muscle fibers are allowed to quickly shorten about I % of /,“. muscle tension will drop from peak isometric force to zero.

Tendon was assumed to be elastic and linear:

Pr=(PJ$).Er; &r=(lr-(,‘)/I,T

=kT.(IT-l;); (A2.4)

k’=(P,Ic~)Jl~.

Since high muscle forces arc dcvcloped during the main propulsive thrust in maxima) jumps, a linear approximation to the well-known nonlinear stress-strain curve (for review. see Zajac, 1989) was believed justifii, and recently this justification has been substantiated through sensitivity studies (Pandy and Zajac, 1989).

Pcnnation. because it is assumed to increase as the muscle fibers contract (Fig 3). affects the relations among tendon. muscle fiber, and actuator force. length. and velocity in the

“... . . . . . . . .-....m ///

Fig. Al. Material propcrtia of muscle tissue. (a) The force-velocity rclarion of the CE specifies the force-velocity property of muscle. The curve intersects the vertical axis at the maximum shortening velocity for muscle (V,), assumed to bc 10 If s- ’ for all muscles. The horizontal intercept denotes the muscle-specific maximum isometric force (P,) that can be developed. Note that the asymptote designating maximum fora generated during lengthening is assumed to be 1.4 PO. (b) The static properties of the PE and CE arc given by ihc force-kngthcu~e of pa*rive and active mu&k, respcctivcly~ The PE is assumed to generate force Pra(lY) at lengths 1” > I,“. Isometric active muscle is assumed to generate an additional muscle force PE (as

given by the CE), which, when fully activated, is the P:f(l”) curve.

I 198 -- M. G. PANDV et al.

following way. Sina

cos I = J[ I -( W/in)*]; W= cons;. (A251

then

PT= P”cosz

lMT=lr+lMcosa (A2.6)

uur = ur + v”/cos 1.

The dynamical equation for musculotcndon contraction dynamics can now be found. Recognixing from Fig. 3 that:

P”=FC+P=; P’ I 1” (A2.7)

pF’= pcz; #=VC’+oCE

and that

dPN dPH dlH -2- .-,k’.“H

dt dl* dt (A2.8)

and defining

k”‘=kMcosa+(PT/lM)~tan2a (A2.9)

it can be shown by combining equations (A2.7). (A2.Q and (A29) that the contraction dynamics for the musculotendon actuator shown in Fig. 3 becomes

dPr (k”“cosa)- k* -1 dr (kY‘cosa)+kT

.[“+J.“CE]. (A2.10)

Sina

k’L~/(lY)=/2(fr,lYTj; uc”~j,[pCc/(P~(ly,o(~))l~o)]

k= =Jj(P3 =f,(Pf. 1”) =4CPr. lMT. awl (AZ.1 I)

then quation (A2.10) can be written ns

df’ (AZ. It)

In many regards, this model for contraction dynamics is a simplified version of the one developed by Hatxe(1977.1978). though similar to those used by others to study motor control [e.g. Winters and Stark, 1985; for review, see Winters and Stark ( 1987) and Zajac (I 989)].

Activation (excitation-contraction) dynamics was also assumed to be tint-order. A bilinear first-order differential equation is used,

(A2.13)

though in this study u(r)=0 or I since the control signals are on-og (see Appendix 3). Thus,

W) ,I==(l/r,,)(I -a); u(f)= I; t,,w = 20 ms

do(r) ~-(l/rr.&,l. --(I); u(r)=Q rraII - 200 mS

a,,, =O.OS (AZ. 14)

which can be written as

$fin. U(I)]. (A2.15)

Notia that the time constants for rise and fall in activation are diRerent. Important to this study is the time constant for rise in activation, sina deactivation of the prime movers is not a major issue in the ground contact phase of maxhnum- height jumping A lower bound on muscle activation a,,. (Hatxe, 1977. 1978) is introduced to cope with problems associated with inverting the force-velocity curve of muscle at low activation levels (Levine et al.. 1989; He. 1988).

APPENDIX 3

Optimal control theory

Some basic definitions of optimal control theory are given below. First, the system Hamiltonian is defined as (Bryson and Ho, 1975):

H(& 1, u)= 1;d+A~B’+~,*P*+i:i (A3. I)

where &,A,. I,,. I. denote the components of the co-state vector associated with the states 0.4 PT. and a respectively, and the prime ( ’ ) indicates the transpose of a vector. The vector of co-states J-CL,. 4, l,h 1.1’ is computed from the adjoint dynamical equations (Bryson and Ho, 1975):

i*‘(t)= _f (A3.2)

where 1*(r) is the vector of co-states corresponding to the optimal controls u*. The boundary condition for equation (A3.2) is:

where I,, denotes that the derivative of performana with respect to the state is evaluated at the final time. The co-state equation is integrated backwards in time from I, to find the complete co-state trajectories (e.g. Fig. IO).

From Pontryagin’s maximum principle (Bryson and Ho, 1975). the optimal solution is found by maximizing the Hamiltonian with respect to the controls u. which, from equation (A3.1). is equivalent to maximizing Q. Sina the equation for activation (excitation-contraction) dynamics is of the form (see equation (A2.13) in Appendix 2):

- = d(f) =I, (a) +/2bb(tX dr

/,(a) >O (A3.4)

then, the optimal controls are given by:

l - I if 12>0

aI - 0 if i:at:,<O

; i=l,8. (A3.5)

Hence, the optimal control solution is ‘bang-bang’. If A,,-0, then the control u, can be anything between zero and one (i.e. the control is ‘singular’) (Bryson and Ho, 1975).

AN OPTIMAL CONTROL MODEL FOR MAXIMUM-HEIGHT HUMAN...

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