Journal of Biomechanics 32 (1999) 87—91

Technical Note

A multi-phase optimal control technique for the simulationof a human vertical jump

T. Spagele!, A. Kistner!, A. Gollhofer",*

! Institute A of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany" Department of Sport Science, University of Stuttgart, 70569 Stuttgart, Germany

Received in final form 19 June 1998

Abstract

A multi-phase optimal control technique is presented that can be used to solve dynamic optimization problems involvingmusculoskeletal systems. The biomechanical model consists of a set of differential equations describing the dynamics of themulti-body system and the generation of the dynamic forces of the human muscles. Within the optimization technique, subintervalscan be defined in which the differential equations are continuous. At the boundaries the dimension of the state- and control vector aswell as the dimension of the right-hand side may change. The problem is solved by a multiple shooting approach which converts theproblem into a non-linear program. The method is applied to simulate a human jump movement. ( 1999 Elsevier Science Ltd. Allrights reserved.

Keywords: Optimal control; Musculoskeletal system; Human movement simulation

1. Introduction

Optimal control theory is used as a framework tostudy the muscular control strategy of the central ner-vous system in multi-joint movements. Optimal control,part of the dynamic optimization, has the advantage tomodel not only the excitation—contraction dynamics ofthe muscles, but also the dynamic behaviour of themulti-body system. In literature, optimal control theoryis applied as a powerful model for the calculation ofmuscle excitations and forces which are essential for anobserved (Davy and Audu, 1987; Spagele, 1997) or anoptimal human movement (Pandy et al., 1990; Spagele etal., 1995). The power of the dynamic optimization tech-nique derives not only from the possibility of modellingthe dynamics of the musculoskeletal system in a complexapproach. It also provides objective criteria for the def-inition of a distinct performance and what the motor taskis attempting to achieve (Zajac and Gordon, 1989).Furthermore, multi-point boundary conditions of a se-lected human movement may also be analysed as well as

*Corresponding author. E-mail: albert.gollhofer@sport.uni-stut-tgart.de

constraint path or pure parameter constraints of thesystem model.

The aim of the study is to apply a multi-phase dynamicoptimization approach to a real human jump movementconsisting of an upward propulsion, an airborne anda landing phase. Specific emphasis was given to a detailedunderstanding of how the central nervous system co-ordinates muscle excitations in order to accelerate anddecelerate body segments of the lower limb for a meas-ured one-legged, vertical jump.

2. Method

2.1. Multi-phase optimal control approach

For the simulation of human movement, it may beuseful to divide the optimal control problem into m sev-eral phases with possibly optimizable phase separationtimes

tj3Mt0(t1(2(tm"tfN. (1)

For each phase j, the musculoskeletal system can berepresented by a set of ordinary differential equationscontaining the equations of motion and the first-order

0021-9290/99/$ — see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S0021-9290(98)00135-3

differentials of the excitation—contraction dynamics ofthe muscles

x5 j"f j (xj (t ), uj (t), pj , t), t3Mtj~1, tjN,

j3M1,2 ,mN, (2)

where xj describes the state vector, uj the vector of con-trol functions and p j the vector of model design para-meters. The control vector uj corresponds to the nor-malized muscle excitations, defined as the quotient of theinstantaneous and the maximal excitations of themuscles. For the prescribed type of movement, the op-timization problem requires the following aspects: (a) thecontrol functions, (b) the model design parameters and (c)the phase separation times of the distinct movementphases to minimise a scalar cost functional I

minuj (t), p j, t0, tj

I , (3)

which may be described by a combination of Mayer andLagrange terms

I"U0 (x1 (t0), p1, t0 )#m+j/1AU j (xj(tj), pj, tj )

#Ptj

tj~Ç

L j (xj (t), uj (t), pj, t) dtB . (4)

In addition to the differential equations (2) the solution isrequired to satisfy separable multi-point boundary con-ditions at the initial time t0 of the movement

w0 (x1 (t0 ), u1(t0), p1, t0)*0 (5)

as well as at the phase time points

wj (xj (tj ), uj (tj), pj, tj)*0, j3M1,2 ,mN, (6)

the pure parameter constraints

jj (pj)*0, j3M1,2 , mN, (7)

and the path constraints of the multi-phase optimal con-trol problem

kj (xj(t), uj (t), pj, t)*0, j3M1,2 ,mN,

t3Mtj~1, tjN. (8)

The subdivision of the entire movement into m inde-pendent phases requires the following conditions at theirrespective phase boundaries

Cxj`1 (tj )

uj`1 (tj )

pj`1 D"lj (xj (tj ), uj(tj ), pj, tj ), j3M1,2 , m!1N.

(9)

Fig. 1. Multiple shooting mesh within one phase and various controlapproximation grids.

The optimal control problem is an infinite dimensionalproblem. In order to get a numerical solution, the controlfunctions have to be fitted by polynomial approxima-tions. The state functions describing the time histories ofthe state values, however, are approximated by solutionsobtained from numerical integration (Jansch et al., 1989).Within a particular interval intermediate starting nodesare introduced for better solution of the ordinary differ-ential equations. The integration starts at each node withnew initial conditions which are formally added to the setof optimizable parameters.

Within a shooting interval, the components of thecontrol vector are approximated by polynomial approxi-mations (e.g., piecewise constant values, piecewise linearfunctions or cubic splines). Here, the control componentsare expressed as a function of the control time nodes andthe control values at these nodes. Both are added to theset of optimizable parameters. With the multiple shoot-ing algorithm, the control continuity conditions and theparameterised control approximations (Fig. 1), the opti-mal control problem is converted into a non-linear pro-gramming problem (NLP). The NLP can be solved usingstandardized sequential quadratic programming algo-rithms. The phase connect conditions and the continuityconditions induced by the multiple shooting algorithm,however, must be satisfied (Jansch et al., 1994).

2.2. Application to human movement simulation

Based on the knowledge of their moment arms thedistinct muscle stimulations produce forces. The result-ant torque’s, however, initiate angular accelerations ne-cessary for the multi-body movement that can be de-scribed by generalized co-ordinates q. The differentialequations of the multi-body system can be generallywritten as (Schiehlen, 1991):

M(q) q#gc (q, q5 , t )"ge (q, q5 , t), (10)

88 T. Spa( gele et al. / Journal of Biomechanics 32 (1999) 87–91

Fig. 2. Measured (dotted line) and calculated (solid line) trajectories of the hip angle u, the knee angle t and the ankle angle m of the vertical jumpmovement.

where M is the matrix of inertia, gc the vector of general-ized gyroscopic forces and ge the vector of generalizedapplied forces. Based on muscle excitations as input andthe knowledge of the system dynamics, it is possible tomodel human movement. The kinetics of the skeleton aswell as the excitation—contraction dynamics of themuscles are determined by a set of differential equationsof the first order for each phase

x5 "

x5p

x5v

x5 c

"

xv

M~1 (xp) (ge(x

p, x

v,x

y, t)!gc(x

p, x

v, t))

! (u, xy)

.

(11)

For state space notation, the vector q of the generalizedco-ordinates is represented by x

p, the vector q5 of the

generalised velocities by xv, and the activation state of

the muscles by the vector xc . The vector ! contains theexcitation functions of the modelled muscles and de-scribes the characteristic properties of all muscles duringactivation.

3. Results

A vertical one-legged jump exercise is investigated inorder to demonstrate a complex application of humanmovement simulation. The entire jump can be dividedinto three different phases. The first phase of the jumpmotion starts from a defined squat position and serves toaccelerate the body in vertical direction. The followingflight phase begins with the take off and ends with theground contact of the foot. The third and last phase ofthe movement describes the deceleration and thestabilisation of the body after landing. The biomechani-cal system of the vertical jump motion consists of a singlelower limb activated by nine muscles or muscle groups.Within the mechanical model, the position of the leg isdefined by the absolute co-ordinates q

1, q

2and q

3which

are used to calculate the trajectories u, t and m of therelative angles of the hip, the knee and the foot. The timehistories of the measured and calculated relative anglesare given in Fig. 2.

The calculated normalized muscle excitations of thesimulated vertical jump movement are summarised inFig. 3.

T. Spa( gele et al. / Journal of Biomechanics 32 (1999) 87–91 89

Fig. 3. Trajectories of the calculated control values of the nine modelled muscle groups necessary for the vertical jump movement.

4. Discussion

The human musculoskeletal system is characterized bynon-linear skeletal and excitation—contraction dynamics,high dimensionality of the state vector and a high num-ber of control functions. Therefore, it is necessary to useefficient and robust dynamic optimization techniques tofind the muscle excitations necessary for a measured oran optimizable human movement. In contrast to gradi-ent-based methods (Miele, 1980; Pandy et al., 1992),parameter optimization algorithms have the advantageto reproduce the dynamical discontinuities realisticallyas observed in human movements. Due to the multipleshooting approach, constraint gradients require pertur-bed trajectories of only one shooting interval and dis-

turbed initial states, parameters and control functionsof the next following interval. This causes a significantreduction of trajectory integration (Jansch et al.,1994). The parameter optimization approach is a robustand efficient method to solve human movement prob-lems despite the computational difficulties. This algo-rithm is widely accepted (Pandy, 1992). Phase-orientedoptimization supports especially the solution of multi-phase human movement problems with discontinuities,changing dimensions and changing boundary conditions.With the developed biomechanical model includingthe dynamics of muscle excitation, it is possible tosimulate and optimize human body motions in order toelucidate the control strategies of musculoskeletalsystems.

90 T. Spa( gele et al. / Journal of Biomechanics 32 (1999) 87–91

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