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aal-929ol90 s3.m+ .oo
Vol. U. No. 12 pp. 11lS1198.19’70
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Pmmedia Great Brilun
t99oRrpmonRorpk
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 serieswith
tendon. Tendon is assumed to be elastic, and its properties an defined by a St-train
curve. The
mechanical behaviorof muscle is descrihcdby a Hill-type contractileelement, 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 ofmass
of the body subject to body-segmental, musculotcndon. and activation dynamics, a xcro verticalground
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*
reproducesthe major featuresof a maximum-heightsquatjump (i.e.limb-segmentalangulardisplacements,
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 progressed 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 demonstrated. 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 Education, 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 expcrimental 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 elucidating 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 decreaseprior 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 suggestionsthat
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 available 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 circumvent 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 proposed by McGhee et al. (1979). However, this approach 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 maximus (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 foursegment 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
1187
control model for jumping
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
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-scgmental parameters arc specified in Appendix I.
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;
LzTis tendon stilTness: kSE and k’” are the stilTnessof the scrieselastic (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
To
neural
torques. The details of equation (I) arc given in
Appendix I.
Muscuhtmdon
Each musculotendon actuator was modeled as a
three-element, lumped-parameter entity (muscle), in
serieswith 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 shortrange 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),:
signal
O<o(r)<l.
(2)
The details of equation (2) are given in Appendix 2.
excitation, u(t)] and muscle
WC have constructed a lirst-order
[mu&
activation [a(f)],
equation:
4l)=(ll~,i..)(l
Bynumics
dynumics
describe the time lapse between the incoming
-“)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 Alexander and Vernon (1975). Woo et al. (1982). and Butler
hi. G. PANDY et al.
I188
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
(i:g)
/:,
Tendon
PNS
Actuator
1:
(m)
s:
(%)
SOL
20.0
0.034
4235
0.360
2.5
OPF
10.0
0.036
3590
0.405
2.6
TA
GAS
VAS
RF
HAMS
GMAX
5.0
12.0
10.0
14.0
9.0
0.0
0.070
0.062
0.090
0.075
0.106
0.182
1400
2370
5400
930
2350
2650
0.265
0.411
0.206
0.323
0.390
0.090
2.7
3.9
3.0
2.6
2.6
5.3
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 differencesmay
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).
normalized isometric
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).
An optimal control model for jumping
dynamics
The dynamical quations for the overall musculotcndinoskelctal system arc therefore:
MusculotrndinoskeIetaI
i9-A(6)‘‘[B(6)V+C(8)+DM(B)Pf+
fi:=A(@, b, P:, aJ
W==(W~)(l
Z-(8,8)] (4)
i= 1.8
-ar)ur+(llrc.u)(a,i.-a,)(l
vertical ground reaction force). We imposed interon the limb-segmentalangles in
order lo prevent joint hyperextension. These, however,
were found to be inactive. The constraints which
remain active are:
mediate constraints
(5)
-uJ
i--1,8.
(6)
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, . . . , u8J’ is
coupled lo muscle activation through excitationcontraction 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).
Obu(t)<
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 maximizes 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:
Ju4 4 t,)= U’, )+ WQYZS
(7)
1
(8)
and
F,(e. 8,3)&=
t
mi(‘i;,+g)
151
=0
(9)
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
OPTIMAL CONTROL PROBLEM
1189
oj the initial states
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).
During the initial squatting position. we hypothesize 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:
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.
J= i V’:lPd2
The constraints, which define the problem, are
the dynamical equations [equations (4)-(6) 1, a set of
inequality constraints which bound the magnitude of
each neural control signal, and a terminal equality
constraint that specifiesthe instant of lift-ofT(i.e. a zero
where Pf is the value of the ith musculotendon force,
and P,,, is the peak isometric strength of the ith muscle.
Note that the parameter P,, is directly proportional
to muscle cross-sectional area. Therefore, the static
optimization problem was to minimize equation (10)
u(t)
Excitat$-CbC~;action
i
a(t)
I
(W
111
Musculotendon
Dynamics
ph
_
Skeletal
Dynamics
e(t).b(1).m
ho
Fig. S. Block diagram showing interactions among the major compartments of the musculotcndinoskeletal model. Note that excitation-contraction dynamics is dccoupicd from musculotcndon and skeletal
dynamics.
M. G. PANDYet al.
II90
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.
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 assessthe 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 assessthe accuracy with which our
algorithm is able to locate a local maximum.
Computation o/the opfimol 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 implemented a modified version of an algorithm developed
by Polak and Mayne (1975) (seeSim (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-
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 recorded 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
An optimal control model for jumping
1191
force (8 body
weight)
Jo0
I
hori:ontal
I
0
20
do
c.2
do
;.v
f/oof 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 suggeststhat 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.
wt ,
0
20
%
,
40
,
60
,
do
I.12
IW
o/ground contact rime
Fig. 6. Limb-segmental angular displacements (solid lines)
and velocities(dottedlines) 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 xgmental angular velocity) increasesjust 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
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) dcmands 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 musclesare not
activated simultaneously, but rather in the above
noted sequence.
While the model successfullyreproduces the major
features of a maximum-height squat jump, shortcomings 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 hypothesize 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 responseof the
model is the increasing limb-segmental angular speeds
M. G. PANDY et al.
1192
muscleforce
tm~b
neural input
(N)
I
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 co-
looo-
n
CMAX
I/ \
I
0
96of prorcnd
conractrime
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-
ordination 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
thesized that limb-segmentalangular speeds decrease t, to a new lift-off time (t,). The height of the new
duringhumanjumping becauseof flexormuscleactiv- optimal jump would also be different. either higher or
ity (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
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
time for VAS [see Fig. I(VAS)]
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 musculotendinoskeletal 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-
An optimal control model for jumping
normalized
1 costates
a
1193
muscle activation
_;;!$,
T
01
i
I’
L
I
I
.mI0
% 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 afTcctsoptimal jump height much more than
the other co-states.
lat displacements therefore have the greatest potential
for improving jump height. This is especially noticcable 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 liftOR [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 musculotendon actuators: SOL, GAS, VAS, and GMAX. Also
shown are the corresponding control signals (heavy
I
I
10
40
60
so
,a,
46of 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 produces 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. PANDYet 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 bodysegmental motions, muscle forces, and muscle activations, but it simultaneously delivers a fully coordinated result (compare the stick figures in Fig. 6 with
those in Fig. 9).
Due to the complexity of our musculotendinoskeletal 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 precision), 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 inaccuracy 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 contidence 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 sequenceof
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 assessthe response of the model
more closely, we have collected experimental results
for several subjects instructed to perform a maximumheight 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-fiberspeedis 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) affectsjumping 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 stiffnessof tendinous structures in
the human triceps surae (Bobbert et al., 1986a. b).
Investigating the effect of ankle plantarticxor tendon
stiffnessamounts 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 momentangle 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
stiffer) than those crossing the ankle (Zajac,
therefore
1989). we
An optimal control model for jumping
assume that overall jumping performance is much less
sensitive lo tendon stitrncss of the more proximal
muscles. By introducing such changes to our model,
we will be able to quantitatively assesswhich factors
dominate performance (i.e. whether jump height is
more significantly influenced by intrinsic muscle-fiber
speed, strength, or ankle-plantarflexor tendon stiffness).The ability to address such global issuesmakes
the use of optimal control theory especially appealing.
Acknowledgements-We thank Eric Topp for assistance with
computer simulations and David Delp for his help wifh
figure preparation. This work was supported by NIH grant
NS17662, the Alfred P. Sloan Foundation, INRIA, and the
Rehabilitation R & D Service, Dept. of Veterans’ Affairs.
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-a,,w4-4)
alI
A(B)=
-a11cdI-4)
a~,cos(4-~,)
-a,4cos(B,
B(e)=
of
I,, = distance of segmentcenter mass from diital end
mr=massofsegtnenti
19,= angle that segment i makeswith the horixoatal.
-e,)
0
-a,*sin(g,-0,)
a,,sit@,-e,)
-a,,sin(l,-8,)
al2
~,,ww4)
-a13=de,-W
-a23W@,-f%)
~l.we~l)
-~,~~~~(e,-e.)
a33
-a,*
sin@, -e,)
0
-o,,sin(e,-0,)
-a*. sin(tJ, -ed)
0
0
D=
where
1
I
-2 0
0
2
2
-2 o-2
0
~,,=I,+m,If,+(m,+m,+m,)l~
~tl-14
01,-c,
1I
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)
1,(m)
Ukg ml)
Foot
Shank
Thigh
HAT
2.2
7.5
15.1s
51.22
0.095
0.274
0.251
0.343
0.175
a435
0.400
0.343
:g
01126
6.814
The moment applied to the foot segment from the nonlinear,
damped. torsional spring is defined as:
I
Body-segmental dynamics
when
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
1
1 1
2
32.826-839.
APPENDIX
c&q cost&
[
-4)
-o,,sin(ll,-8.)
a,,sin(g,-e.)
-a,. sit@, - B.)
0
-c,gcose,
cs c0se,
-cacose,
a@‘=
-0.)
k
-a,,sin(B,-0,)
-a,,sin(8,-8,)
0
-o,,sin(tJ,-8,)
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442-444. Minneapolis, MN.
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.
-a,.d4
a,.we.-e,)
-a,.wh
otherwise
F(6, 6)=0
e, c 0,;
An optimal control model for jumping
where:
k,= spring stiffnessconstant (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 musculotendonmodel
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.
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$tY).a(t),
where PT,L(I”) is th;
force-kngth curve of active muscle when dt)= I [i.e. when
muscle k fully activated; Fig AI(b)]. The fo~vclocity
relation of muscle is accounted for by assuming that the force
FE generated by the CE is agected by its velocity rec. and is
given by a pr(ucr)
relation [or, quivakntly.
by a
velocity-force relation; see the ucr(~‘) curve in Fig Al(a)].
The force generated by the CE was assumed to be
F&=(P:,CII”.
P, = peak isometric active muscle force
If = muscle fiber length where P,, is developed
a = muscle fiber pennation when P’ = I,”
/~-tendon slack length
eX = tendon strain when P’= P,.
Maximum intrinsic shortening velocity was assumed to bc
o(t)J/P&
=(P:.s(f”)..(r)/P,).
F”co’“)
prL(oCE)
UC&
-/!
c~E/v%wo)l
(A2.2)
whcref,( *) is the velocity-force curve in Fig. Al. Stiffnessof
active musck (the SE) was assumed to be
kS”=(loOP=+
loP,)/I;.
(A2.3)
This stiffnessimplies that when muscle fibers are allowed to
quickly shorten about I % of /,“. muscle tension will drop
from peak isometric force to zero.
Tendon was assumed to be elastic and linear:
Pr=(PJ$).Er;
=kT.(IT-l;);
&r=(lr-(,‘)/I,T
k’=(P,Ic~)Jl~.
(A2.4)
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
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.
of
(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.
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:
119-l
I 198
--
M. G.
PANDV
et al.
which can be written as
following way. Sina
cos I = J[ I -( W/in)*];
W= cons;.
$fin.
(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”
pF’= pcz;
#=VC’+oCE
(A2.7)
APPENDIX 3
dPN
-2dt
dPH dlH
.-,k’.“H
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
(k”“cosa)- k*
(A2.10)
.[“+J.“CE].
(kY‘cosa)+kT
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
i*‘(t)=
k’L~/(lY)=/2(fr,lYTj;
(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):
Sina
_f
(A3.2)
uc”~j,[pCc/(P~(ly,o(~))l~o)]
k= =Jj(P3 =f,(Pf.
then quation
(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 maxhnumheight 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).
and that
dPr
-1
dr
U(I)].
1”)
=4CPr.
lMT. awl
(AZ.1 I)
(A2.10) can be written ns
df’
where 1*(r) is the vector of co-states corresponding to the
optimal controls u*. The boundary condition for equation
(A3.2) is:
(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,
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):
(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
do(r)
~-(l/rr.&,l.
-
dr
= d(f)=I,
(a) +/2bb(tX
/,(a) >O
(A3.4)
then, the optimal controls are given by:
-a); u(f)= I;
--(I); u(r)=Q
t,,w = 20 ms
l aI -
rraII - 200 mS
a,,, =O.OS
(AZ. 14)
I
if
12>0
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).