DEPENDENCE OF JUMPING PERFORMANCE ON MUSCLE PROPERTIES WHEN HUMANS USE ONLY CALF MUSCLES FOR PROPULSION FELIX

E. ZAJAC

Mechanical Enginrering Department, Stanford University. Stanford, CA V-1305. U.S.:\. and Rehabilitation Research and Development Center, Veterans Administration hledrcal Center, Palo Alro. CA 94304, U.S.A.

ROGER W. WICKE Rehabilitation

Research and Development

Center, Veterans Administration CA 94304, U.S.A.

MedIcal Center.

Palo Alto.

and

Electrical

Engineering

WILLIAM

S. LEVINE

Universrty

of Maryland.

Department,

College Park. MD

20742. U.S.A.

Abstract-Using optimal control :echniques. maximum height jumps were simulated for humans who held their body rigid except for the ankle. Three dynamic models ofankle torque generation based on known call muscle properties were used. Force and kinematics obtained from the simulations using nominal and perturbed parameters were compared with data obtained from humans who had performed this type ofjump. One torque model incorporated the series elastic, force-length and force- velocity properties of mu&. Our results suggest that higher jumps would be achieved by those who have the most compliant and fastest contracting muscles. It was also.found that height attained depended much more on the ability of muscle> to generate isometric force at long lengths than at short lengths. Studies of forward and strictly vertical jumps using similar computer methods suggest that for any maximal jump the optimal strategy is first to achieve a unique state (position, velocity and accelerationt H rth the feet flat on the ground, and then to maximally activate one’s calf muscles until lift-elf. ISTRODUCTION

strategy and the most sensitive. observable features of this type of jump

In previous papers. we have described our analytic and computational

method

to find the optimal

that should be used by a jumper maximum propulsion.

attempting

strategy

maximum

height jump

the muscle

Relevant

has

to this study is the fact

velocity and acceleration activation

of the calf

should then be increased to maximum there until the jumper

desirable

phase of propulsion

of the

Our

muscles

the airborne and Trinkle,

which

began

with

showed

that

minimal

simulated

The studies reported here are intended to determine

must be included.

interpret

this model in terms of known

Wilkie,

even though

1956)

and

system

characteristicsand

to

muscle strucdynamics also,

muscle properties

includes

with

it is difficult

(Hill,

series elasticity. insertion-origin

We found that this model, though slightly

more complex

than the others, provides good agree-

ment with experiment muscular control 513

In this paper we

first order

coethcients,

contractileelement

1983.

the

FC curves which are comparable

by using a simple,

geometry.

only

Fr curves (Levine cr ul.. 1979); thus.

ture. Our third model has first-order

how changes in muscle model structure will affect the

a zero-order

incorporates

constant

1938;

Nocemher

models

but it is based on well-known

1982).

Receirerl Auytcsr 1983; in reriserlfbtm

(see Methods)

characteristic of muscle, is insuthcient for

muscle dynamics

phase, so this

phase was not of interest to us. However,

results

to the experimental

height

phase is of interest to others (Hubbard

muscle

model,

producing

at lift-off (when the toes leave

of the center of mass in the airborne

use

previous

force-length

and maintained

the maximum

involved in solving the optimal control

to

muscle

which we examine in this paper.

the ground) uniquely determines

to faithfully

complexity.

leaves the ground. It is this last

The state of the jumper

be in order

and increase for higher orders of system dynamics. it is

upper body must have a specific value. Once this heeloff state is reached,

must

problem depend on human interaction

phase is unique; the heels must be on the ground, and position,

model

computations

that the state of the jumper at the beginning of the last the angular

One

reproduce the essential features of the jump. Since the

been shown to consist of three phases (Levine rr al., 1983; see Methods).

jump).

more natural jumps we desire to know how complex

to reach a

height when only calf muscles are used for The optimal

(called a two-segment

reason for this interest is that for our future studies of

and gives insight into neuro-

of jumping.

F. E. Z~J~C. R. W. WICKE and W. S. Ltvl~

514

Maximum forward jumps and maximum vertical jumps (i.e. maximum height jumps with no horizontal velocity allowed at lift-off) were studied in order to see how the optimal strategies for these jumps differ from those for a maximum height jump. It would seem that the strategy used in a maximum vertical jump should be similar to that used in the maximum height jump, since in both cases most of the total available energy should be channeled into vertical rather than horizontal propulsion. Computational difficulties were encountered in solving the maximum vertical jump optimal control problem. Even so, our results indicate that the optimal strategy for any type of jump is to achieve a heel-off state unique to the type of jump being performed, in which the body is pointed approximately in the direction of desired motion, immediately followed by maximum activation of the calf muscles until body liftOff. METHODS Esperimrnts

usiny human

subjects

Data was collected from three young adult males of about the same age (24 yr) and physique. The experimental records shown in this peper were obtained from one of these subjtrrts and are typical of those from the other two. Data collt%tion procedures and algorithms used for determining kinematics and kinetic properties are described elsewhere (Levine et ul., 1983). Briefly, each subjcxt was instructed to jump as high as possible from a Kistler force plate and keep his entire body straight and rigid during propulsion, except for his feet and ankles. Ideally. joint motion should have occurred only at the ankles; however, the subjects were wearing tennis shoes, which probably allowed some motion within the foot. A S&pot system was used to record the trajectories of the joints, and from this data, kinematic information was obtained. Angular velocity was derived by differentiating the sequence of joint positions. Anthropomorphic data was obtained by direct measurements and regression equations. The force-plate provided records of vertical and horizontal reaction forceand center of pressure. Computed torque about the ankle and EMG records are not shown here but are reported elsewhere (Levine et al., 1983). Referring to the two-segment model in Fig. 1, data obtained from these jumps included tit, UJ,, U,. tu,; F,, the vertical reaction force; and F*, the horizontal reaction force along the antero-posterior axis. lnrroduction

to merhorlsjor simulurion

Fig. I. Model of the two-segment jumper. m, and m, are the masses of the foot and body segments. located at the respective center of mass ofeach segment. Torque is produced about the ankle by the calf muscles (assumed to act in concert) of length, f.. Measurements of the human subject who performed the jumping task were used to estimate the parameters of the model: I, = 0.0950 m, 1, = 0.080 m. I, = I.1 I m; m, = 2.2 kg, m2 = 73.87 kg, U, = 34.31’. U,,, = 90. deg; r, = 0.12 m, c2 = 0.435 m. The moment arm, r. is assumed to be a constant (= 0.05 m.) /, and I*, the moments of inertia about each segment center of mass are 0.008 and 12.072 kgm’. respectively. w, i 0, and ojL i U,. Fh and F, act at the toes in the x- and y-direction.

simulation to study maximum height jumps. Each muscle model used in this study assumes that all plantar flexor muscles (soleus, gastrocnemius, tibialis posterior, peroneus group, flexor digitorum longusand flexor hallucis longus)can be combined into one equivalent muscle producing ankle torque. M us& mot/r/s A und B Muscle model A (set Fig. 2) is a linear first order lag actuator followed by the torque--angle characteristic of the

A

sfudies

Experimental jumps and computer simulations were studied in order to develop a reasonable model for maximum height two-segment jumps. Assuming a maximum height performance criterion, we wanted to see how variations in the muscle model structure and in the isometric torque-angle characteristic would affect the behavior of the model relative to the experimental jump. Also, we modified the model to study theoretically how the neuromuscular control strategy should change in order to perform two other types ofjumps: a maximum forward jump and a maximum height vertical jump, for which no horizontal velocity of the center of mass is allowed as the jumper becomes airborne. This latter jump will be referred to as a vertical jump in order to distinguish it from the maximum height jump for which horizontal velocity is not constrained. The methods described here focus on the alterations to our basic simulation algorithm which is described in previous papers (Levine et al., 1983). Three different muscle models were used in the computer

Fig. 2. Block diagram of muscle models A and 6. T,(U) is the isometric torque t’s ankle angle curve. r is the time constant for isometric contractions.

Jumps

ankle planter-flexors. The model .4 and B is that

ustng only ankle

propulsion

515

only difference between muscle in model B, the torque-angle

characteristtc

multipiles the control. u (muscle activation). before the dynamics act upon it. If the time constant r were due only to mechanical dynamics. model E would more accurately represent the dynamics. Likewise. if the time constant were due only to biochemical and electrical dynamics. then model A would be more accurate. Simulations were obtained using both models A and B with the same ttme constant. in order to compare the ability of each mods1 to represent the total dynamics. However, other considerations will complicate this simple interpretarton, as HC shall see

Model C more accurately incorporates known physiological dynamics and geometry of the musculoskeletal system. The model for muscle force development is a two-component version of Hill’s model (Hill, 1938; Wilkie, 1956)and consists of elasticity in series with the contractile element. Passive parallel elasticity can be Ignored in this study since the range ofankle motion in the two-segment jump is restricted to that corresponding to shorter muscle lengths. in contrast to eyemovement muscle models for which paratIo elasticity is important (Clark and Stark. 1974; Lehman and Stark, 1980; Lehman, 1982). Assume that the series elastic element obeys Hooke’s Law P = !i (x, - I,,,). (1) whcrc P = muscle force; L = muscle and tendon stiffness; x, = elastic element length: x,, = resting length of the elastic clement, and that the force velocity relationship of muscle can be linearized (Bahillc~ al.. 1980; Lehman and Stark, 1980) and parameter&d for each muscle length (Abbott and Wilkie, 1953; Marsumoro. 1967a: Bahler Ed nl.. 1968: Clark and Stark, 1974) and level of activation (Jewcll and Wilkic. 1960) such that P-

P,(L)U

xi=---

(2)

b

where s, = lengthening velocity of the contractile element; P,(L) = isometric force-length curve of muscle; b = damping of muscle; u = activation level of muscle. Recognizing that I!_ = I,. f .X 0;

(14)

(31 control constraint. OSU 0

(16) and

0, > 34.31

For maximum height jumps, the solution to this optimal control problem has been found to consist of three parts (Levine rr a/., 1983). For clarity. the solution is presented here. In phase I the two-segment jumper should fall forward. using any one of a variety of muscle activation patterns, to an equilibrium state in which the center of gravity is directly above the toes. In phase II. with the heels still on the ground, the jumper should move his top segment forward (in most cases) with the maximum acceleration possible without allowing his heels to come 01Tthe ground (i.e. the control is a feedback control which maintains the center of pressureat the toes). Such a trajectory requires a unique torque to be generated at each instant by using a control, u(r) < I. This trajectory will be referred to as the phase II trajectory, and has been found to lie on the boundary of states in the (0,. oz) plane which are reachable with the heels on the ground. When the jumper reaches a specific state along the phase II trajectory, maximum activation (u = 1) should begin and last throughout the third phase of the jump. This tinal phase (phase III) terminates when the jumper leaves the ground, that is, when the constraint as given by equation (14) is violated. In order to achieve the optimal jump, the state at

(17)

is violated. However, thisconstraint was never vtolated for the range of parameters used in this study. Intcrucril-r

compuler

ulgoriihm

IO/~

muslmum

jumps

To find the optimal solution for each maximum jump, the set of reachable states with the heel on the ground was searched to find the state yielding maximum performance. This was achieved by specifying a reachable state (Us, w,) as the initial state for phase III of the jump, and then integrating the equation of motion forward in time using a Runge-Kutta approximation scheme. After forward integration, thecostate equation was integrated backward in time from the terminal costate. This provided an additional check for the optimal solution; the control Hamiltonian for the optimal solution should equal zero for all time. For maximum height jumps. the optima) initial states were always on the phase II trajectory. The phase II trajectory lies on the boundary of the set of reachable states with the heel on the ground. In order to verify that the optimal solution was on the boundary. adjacent points within the region of reachable states were also checked. The major difference between a maximum height and a maximum ioruard jump is a ditference in the performance criterion. See Appendix A for the derivation of the pcrformante criterion for the forward jump. The vertical jump’s performance criterion is the same as that for the maximum height jump. However, for the vertical jump it is necessary to add theconstraint that the horizontal velocity ofthecenter of mass at the terminal time (lift-off) is equal to zero; this involved changing the formula for determining the terminal costate.

We made the assumption that the toes are always contacting the ground during the propulsion phase. The terminal time is detined to be that time at which the body leaves the ground (when the dynamic constraint, equation (14). is violated). Oprimul jumpiny sfrcrreyj

1

dL

I rhi‘ P-c!lrbz+T u = ~ TJUI i k,(T+k,,)

RESLLTS

The stick figures in Fig. 4 show how the body should move after heel-off during

the maximum

height and

forward jumps using muscle model C with the nominal parameter

set. Notice that after heel-off the maximum

height jump rotation

(Fig. 4A) consists of a monotonic forward

of the foot while the upper

body remains

relatively motionless with respect to the ankle. From a vertical orientation

and prior to heel-off (not shown),

the upper segment needs to fall forward only slightly in order to achieve the optimal The maximum

forward

heel-off state.

jump,

on the orher hand,

requires the upper body to continue to fall forward to a relatively

small angle prior to heel-off. After heel-off

body should continue to fall at a relatively high angular velocity as the torque generated about the ankles propels the body forward. While this (Fig. 4B). the upper

jump

is highly artificial

representative indicate

and would

probably

not be

of a forward jump by a human, it does

the difference

in strategy between the mani-

Jumps usmg only ankle propulsion

.

.

.

517

using model A tend to vary more than those using model B. For models A and B r,(O) curves were altered from the nominal curve as shown in Fig. 5. These other

l

A & 1

’

Tafe)

c&Lib

0

50

100

200_

(N-m)

Yyip \\ ~\ \ ., ..\ ‘\ ‘q\ \ l..\

\ .\\ “\ ‘, ‘:\\

i

15Oms

Iliftoff at 160 ms)

8=0,+8,

0

50

loo

150

idegi

ms

ll~ftoff at 163 ms) Fig. 4. Stick rcpresentatlons of the maximum height (A) and forwtrd (B) jumps. Elapsed time after heel-otf is shown beneath each stick figure.

Fv (Nl

height jump and a maximum forward jump. However, this might be an approximation to a dive.

mum

show the results of changing various model parameters for each of the three muscle models. These results are discussed in detail below. The vertical reaction force, F,. was chosen as the primary variable to plot because it is easily measured in theexperimental situation and is closely related to the instantaneous torque produced about the ankle joints. Also, F, is more sensitive to changes in model parameters and structure than the kinematic variables, which are relatively insensitive to these changes (Levine rr al., 1983). Thus, F, can be used to deduce qualitative and quantitative changes in both muscle force and torque. Kinematic data was. as expected, found to be rather insensitive to model structure and parameters. For all simulations. II, monotonically increased until lift-off. and U2 remained relatively constant throughout all maximum height jumps (e.g. Figs %I. D). One major ditference between the experimentally obtained curve and all simulated curves is that the experimental curve shows a plateau in the value of 0, just prior to liftofT [Fig. 5C. solid curve). Figures

Comparison

S-1 5

of rrsuks from

models

A and B

The most salient difference between models A and B is that peak vertical force occurs nearer the end of the jump using model B. Compare Fig. 5 with 6, and Fig. 7 with 8. For the same variation in parameters, the duration of the jump. which we define as the time between heel-off and lift-off, varies much more using model A than model B. Also, the peaks of the F,. curves

C

60

0, (deg)

1 D 80--

........-

8, Meg J 60 00.7

0.8

0.9

1.0

Fig. 5. Simulated jumping results using model .4 ior various torque-angle characteristics (A). Curve c is the nominal 7,(O) curve. No jump is possible for curve cr. (B). tC) and (D) show the vertical force F, and the U!, U2trajectories corresponding to each 7,(U)curve shown in (A). The experimental data is also shown (solid lines). Modified performance measures P’ (equation 18) are as follows: 6 0.0723 m: c 0.0762: d 0.0806; rO.OSSO. This measure represents the increase in center of mass height above its position when the jumper rs standing at rest.

518

F. E. ZAJK. R. W. WICKE and W. S. LE\?.E

curves were obtained by multiplying the nominal curve (without saturation) by a constant and then applying the saturation value of 340 Nm. This is equivalent to a shift with respect to 0. and has the effect of changing the torque available at those values of U which occur toward the end of a maximum height jump (plantarflexion). Notice that for model A, a jump does not occur when ankle torque capability is suhiciently limited (curve a, Fig. 5). For model A, the most obvious effect of shifting the T,(6) curve to the right is to increase performance, or height jumped (Fig. 5). For all figures in which computer simulations are shown, modified performance measures, P’ = p - r;. (IS) nt,iz

yi =

nt2 I,

(Ii+(nJ1 >s’nu,+(nJ, +nJ,)

+mz)

are listed in the figure legends. This measure P represents the increase in center of mass height above its position when thejumper isstandingat rest with his feet flat on the ground (i.e. 0, = 0, and 0, = 90‘) and allows meaningful comparison of performance bctween jumps for which segment lengths are changed. For model B, performance also increases for similar torquecurve shifts (Fig. 6). As with model A, for model B the peak value of F,. increases with the shift, but propulsion duration is affected little in contrast with the large decrease occurring in model A. An increase of T resulted in an increase in the total duration of the jump (Figs 7 and 8). However, duration increases were not as great for model B, indicating that this model is less sensitive to changes in T. Using model A, the peak torque occurs closer to lift-OK time as r is decreased. However, just the opposite happens using model B. As T decreases. the time between peak torque and lift-off increases, although the e!fect is small. Performance increased for both models as r was decreased. To conclude, based on the above results, it should be apparent that performance and trajectories are more

Fv (N)

Fig. 7. Simulated jumping results using model A for various time constants r. T = 0.100s is the nominal. No jump is possible for T > 0.125 s. Corresponding P’ values for T from

high to low are 0.0741.0.0762. 0.0855 and 0.0939 m.

Fv (N)

o-

0.7

I

0.8

0.9

1.0 s

Fig. 8. Simulation results using model B for ditferent I. P’ values for T from high to low are = 0.130, 0.134, 0.142 and 0.139 m.

when model A is used. Notice from Fig. 9 that an increase in I, enhances the

sensitive to parameter changes

modified performance measure P’. The reason for this enhancement of net airborne height is that with longer

-.e

Fv

hJ

(N)

(N) 800 -

0.7 Fig. 6. Similar to Fig. 5 using muscle model B. Only F, curves for T,(U)characteristics a. c and e are plotted. P’ values for ~1.c and e are 0.121. 0.134 and 0.141 m.

0.8

0.9

1.0 s

Fig. 9. Simulated jumping results using model B for different 1, _Corresponding P’ values for I, from high to low (nominal = 0.0950mJ are 0.171. 0.154. 0.134 and O.IlOm.

519

Jump> using only ankle propulsion

We found that an mcrrase in the dampingcoelhcient

1600-1

reduces both the peak force attained

during

propul-

sion (Fig. 1 I) and the height reached while airborne. Even though propulsion the nominal

ing coellicient. FV

duration

is prolonged

I50 ms to 200 ms by doubling this prolongation

from

the damp-

is not enough

to

compensate for the reduction in force. To understand

(N)

these results it is useful to recognize that the parameter h appears only in the second term of equation can be considered

to similarly

(10) and

atTect dynamics as the

time constant T in models A and B. (Compare

Fig. I

I

with Figs 7 and 8.)

0.7

0.8

0.9

If the muscle (tendon) stitfness parameter

1.0 s

Fig 10. Simulated jumping results using model A for different varying I,. P’ values for I, from high to low (nominal = I.1 I m) are 0.0750, 0.076, 0.0777 and 0.0795 m.

feet the upper body can fall further forward during the first phase when the feet are flat on the ground. Thus the calf muscles, being at a longer length at heel-OK can contract

afterwards

in a region

of the force-length

curve that generates more work output. In contrast,

notice from

Fig. 10 that P’ decreases

as I, increases. The explanation

above is again valid. In

this case, a tall person cannot fall as far forward

as a

short person and, everything else equal, the tall person will

not

be able

However, outweighed

to generate

this diminution

as much

propulsion.

in net airborn

height

is

by the higher standing height of the tall

person such that the absolute

height of the jump

indeed higher, at least for the parameters

is

used in this

study.

the

height would increase. even though peak

more compliant

muscle (or tendon), a longer proput-

sion phase compensates force.

In contrast,

longer propulsion the reduction

parameters

(h. 6,. Ii,,,,)= (16000,60,50) jump

in

model

and the nominal

the F, curve

is very similar

occurring

C,

torquein the

to the experimental

curve. However, the simulated jump is still significantly faster than the experimental

jump

(Fig, I I; compare

dotted with solid curve).

the

in peak force. It should be mentioned

value

did reach a maximum

(k, = 5 m-‘).

Similar,

smaller, ctl’ccts on performance, duration

in peak

is increased

phase does not fully compensate for

that performance realistic

for the reduction

when damping

at an un-

though

much

force. and propulsion

were found for changes in k,, (Fig. 13).

To understand

why performance

is enhanced when

muscle stillness is decreased, it is important nize that equation

model C has qualitatively simple lirst-order

to recog-

(IO) has two terms, indicating ditferent

behavior

that

from a

system (e.g. models A and B). Notice

that the parameters

L, and I;,,, appear in both terms

and, therefore. it is useful to compare the magnitude of becomes dominant

nominal

angle characteristic, computed

that jumping

vertical force would be less (Fig. 12). Notice that for a

both terms ofequation

R~w~l~sfiorn model C Using

I;, should

decrease from the nominal. our computer results show

(IO) (Fig. 14). The w (first) term in the latter

part

of the jump.

Because this term is negative, if its magnitudecould

be

reduced, torque in the latter part of the jump would not decline as rapidly. A slower decline of torque near the end of propulsion

is desirable because at this time

the ankle has plantarflexed generating

potential

to a point where the force-

of the calf muscles is small (see

T,(U) curve), and. therefore,

the second term in equa-

1600 1200

Fv (N)

Fv (N)

800 400

0' 0.7 Fig. II. Simulated jumping results using model C for two damping coethcients b. A jump is not possible for b = 24ooO N sm- ‘. P’ values for b = 16000 (nominal) and 8000 are 0.067 and 0.080 m.

0.8

0.9

1.0 s

Fig. 12. Simulated jumping results using model C for three k, parameters. P’ values for b, = 30, 60 (nominal) and 90 rn- ’ are 0.080. 0067 and 0.058 m.

520

F. E. ZAJ.AC.R. W. WICKE and W. S. LEVISE

A

_

1200 Fv (NI

800 400

100

120

140

160

8=8,+9,(deg) B 1600

Fig. 13. Simulated jumping results using model C for three k,, parameters. P’ values for k,, = 25, 50 (nominal) and 75 N m are 0.07 150.0672and 0.0636 m. tion (10) is small (Fig. 14; see curve labeled TDZ). Thus, decreasing muscle stiffness enables developed ankle torque to be maintained for a longer time, and performance increases. As with models A and B, computed results using model C show that when isometric torque remains high into full plantarllexion, performance is enhanced (Fig. 15). However, for curves /. g and 11,both the performance and the F, curves differ only slightly, indicating that isometric torque capability in the ankle angle range greater than about 127” is relatively unimportant to the jump. On the other hand, isometric torque capability at ankle angles less than 127,’ is critical to propulsion, as shown by comparing curve L (for which a jump occurs) to curve i (for which a jump is impossible). Results of maximum

urrrical

jump

On the boundary of the set of reachable states with the feet on the ground, the horizontal velocity at lift-off reverses at approximately (U,, ol) = (82.6’, 0.42’ s- ‘). However, it was impossible to compute the exact reversal point because the horizontal velocity at lift-off is extremely sensitive to heel-off state near this point. This is expected because the system is unstable; that is, the system is an inverted pendulum. Consequently, it is practically impossible to perform a strictly vertical jump using only ankles for propulsion.

I

”

2i

TDZ

Fig. 14. The two components of equation (IO). 7D1 and TD2 are the first and second terms of the equation. 7D1. which is the term containing w. becomes very significant toward the end of the jump.

F"

(N)

800

Fig. 15. Simulated jumping results using model C for different r,(U) characteristics (a). Curve c is the nominal. Curve i is from Sale YI al. (1982) and a jump is not possible. P’ values for curves c. h. y andfare 0.0670,0.0835.0.0882 and 0.0883 m.

Results of maximal

jixwwd

jump

For this jump, the jumper is required to move his center of mass as far forward as possible before landing. The performance criterion for this jump is developed in Appendix A. Using the computerinteractive search procedure, the optimal heel-off state was found to be (U, = 30.80”. w2 = - 142.83” s-r). A diagram of this jump from heel-OF to lift-off may be seen in Fig. 4. It is clear from the diagram that if a person were to jump in this manner. he would fall to the ground after landing because of the low angle and high angular velocity at lift-off.

DISCUSSION

Although there are obvious differences between the maximum height and maximum forward jump, each requires the upper body to attain a specific orientation, velocity and acceleration prior to maximal activation of the calf muscles. One interpretation is that the body must be ‘pointed’ in the optimal direction so that propulsion provided by the calf muscles is utilized appropriately (eg. Fig. 4). One major difference between all of the simulated jumps and the experimental jump is that the experimental 0, trajectory flattens prior to lift-off, whereas Or continues to increase until lift-off for all of the simulated jumps. Since the experimental jumper was

521

Jumps using only ankle propuhon

wearing

tennis

shoes, it is possible

plantarflexion known

from

previous

ceases and tibialis off

(Levine

would ankle

ation

plantarflexion. jumping

occurs

after

to lift-OK

(unpublished

When

ankle

in

muscle

slowing

down

toes are consid-

after

ankle

motion

and

prior

why

an

activation

much less to propulsion

being

it (and other

dorsiflexors)

flexor

produce

activity

activity

is

of flexor

this

would

same

and

torque

not change

to make

it more

consistent

of muscle dynamics.

the

model

A or B is to represent

with biochemical uscular

appropriate.

then

However,

properties

model

activation

torque

neural

inhibitory

impossible

are

with the viscoelastic

reduced

of 0.025 s

comparable constants

to the reported (Gottlieb

However,

and Agarwal,

the simulations

obtained

experimental

than

jump

torque-angle

characteristic

force

parameters, which

model

A.

the isometric the

spect, though,

With

the experimental

model

control

model C also

effect of the transformation

to torque.

of increasing

B

from

However,

the proper jump

Because of this, one may be tempted value

model

the

both model C and A produce

match

time

from

multiplies

before it acts on the dynamics. muscle

both to be

further

B in that

the model

C incorporates

of

choice

of

simulations

reasonably

suggests

the upper

that body

rigid

to perform.

of

may be

Indeed,

to perform

torque,

jump

the task

some

the jump.

torque-angle

characteristic

it is evident

that

the most

the

In retro-

basic features

50”,,

from

jump

began

at an angle

about jump factor

began at II5 which

torque

at

the

torque

of

and

and the 160..

importance

angles

of

is that

the

to be maintained

the isometric

computer

by humans would

with

torque

at

at angles

result was that the higher

more compliant

be accompanied

force during reduction

duration

height. However, height

of using force impulse

are fast and strong. is

plantarflexion.

calcu-

curve rather than peak force alone to estimate should

the requirenient

them to have compliant more

of

of propulsion

the jumping

factors other than force impulse

at lift-OK)

To conclude,

in peak

The compensation

by prolongation

lated from the force-time or propulsion

muscles/ten-

by a reduction

propulsion.

stresses the importance

lexion

The

115’

127’ is low.

this force

body

the

larger

in model Callows

One interesting

vertical

of

and ended at about

reduces

high levels even though

jumps

is already

maximum.

140’ by the end of the jump,

simulated

beyond

its

tendons Calf

for humans

to jump as is for

and calf muscles that

muscle

important

(e.g.

also be considered.

high as possible using only ankles for propulsion

well.

to question

complexity.

torque-angle

a simulated

about

Another

dons

from

simulations

to model

includes thegeometrical

for

constant

1971).

and qualitatively

C is similar

in a

values of muscular

were quantitatively Model

a limit as

e’l (II., 1979). Thus, time

15). this

keeping

experimental

dynamics

approaches

the nominal

make

the isometric

by

reached

1969). In

T

A and B, a value for

(Levine

in the isometric

torque

zero, and these results can be found

paper

may be due to a

less than 127’. At 127‘. the isometric

be more

jump.

we chose

velocities

tif

the reduced

of neurom-

experimental

models

that

is at ankle angles

isometric

previous

to enhance (Caiozzo

crucial range for torque development

time constants

Performance

at slower

the nominal

which are much faster and higher than the

approaches

for muscles

was found

YI ~1.. believe

some people

By varying

less than would

at slower velocities

find it impossible

from

during

mechanism.

(Fig. for

in for

s- ‘, the in

less than 200

of the ankle

while

was apparent

(1938) relationship

Since slight reductions characteristic

difficulty

(less than 144’ s- ‘). They

Since training

performance

quite

contractions

is significantly

by Hill’s

maximum

yields jumps T

matched

associated

of large limb muscles (e.g. Taylor,

fact, for both models

in

jumps

knee extensor

torque

maximal

In spite of the

data. Perrine and Edgerton

that at velocities

Caiozzo

people

existing

A would

much faster than those associated properties

with

the dynamics

and electrical

activation,

structure

If the time constant

the simulated

(I/., 19Sl),

difficult

theories

data.

the b parameter

were not exerting

contractions

~.ico maximum

The three muscle models used in this study represent the model

which

experimental

range of motion.

maximal

slow isokinetic

jumping

we altered

and doing

parameters

pith

more than 500 ms. This difficulty

this study.

in order

and so was deleted from

one model

(1978) have shown that subjects experience maintaining

maximum

is allowed,

if flexor

combinations

in which

error,

tested in isolation.

the boundary

(muscle)

However,

could

height of the jump a progression

experiments

be predicted

the feet flat on the

one control

possible

im-

phase II. During

states with

is unique.

active

circumof limit-

error in determining

over the entire

also found

anterior

to heel-offduring

of the set of reachable

This

than and the

little.

tibialis

and if only

0,

because the weak

phase II the upper segment travelsalong

trajectory.

Fugl-Meyer’s etrort

immediatel)

experimental

However,

to lift-off,

activity

The estimation

to fortuitous the dangers

model

yield the best correspondence

the

may be active prior

many

for

explain

to

extensor

to only

well with the experimental

F,. curve to be affected

allowed,

study

searching

estimation

contribute

this control

but

illustrates

in

reasonable

for model C may be due to the fact that the subjects in

since toe

the strong calf muscles, WCexpect performance

ground,

ing a simulation nothing

which are iacking

,+l produced

be attributed

Such

has a plateau.

mediately

should

stances. This example

observations).

would

addition

and architecture

rl. The fact that model

in cats (Zajac t’l trl.. 19SO) and humans

trajectory

In

activ-

plantar-

anterior,

model

simulations

to lift-

may require

by tibialis

sequence

toes probably

of

strategy

has been stopped

motion prior

changes

the effect

of the toe plantarflexors

fiexion

it is

becomes active prior

1983). Such have

ered, the optimal

muscle dynamics

significant

In addition,

research that calf r_luscle EMG

anterior

ef al.,

activation ongoing

that

of the toes occurred.

strength than

in dorsif-

strength

in

522

F. E. ZAJAC. R. W. WICKE and W. S. LEVIXE

Ac~noHlrdgrmmts-This work was supported by NIH grant NS 17622 and the Veterans Administration. The tvoine assistance of R. DeMaio and P. Polen is appreciated:‘Th; experiments using human subjects were conducted in Professor S. Grillner’s laboratory in Stockholm with the assistanceof Drs M. Zomlefer and H. Carlson.

REFERENCES Abbott, 8. C. and Wilkie, D. R. (1953) The relation between velocity of shortening and the tension-length curve of skeletal muscle. J. Physiol.. Lend. 120. 214-223. Agarwal, G. C. and Gottlieb, G. L. (1977) Compliance of the human ankle joint. J. biorntxh Enyng. 99, 166170. Bahill. A. T.. Latimer, J. R. and Troost, B. T. (1980) Linear homeomorphic model for human movement. IEEE Truns. biomrd. Enyny. B%lE-27. 631439. Bahler, A. S., Fales, J. T. and Zierler, K. L. (1968) The dynamic properties of mammalian skeletal muscle. J. gm. PhJsiol. 51, 369-384. Caiozzo, V. J., Perrine, J. J. and Edgerton, V. R. (1981) Training-induced alterations of the in cico force-velocity relationship of human muscle. J. uppl. Physiot. 51,75&754. Clark. M. R. and Stark. L. (1974) Control of human eye movements: I. Modelling of extraocular muscle. Murhl Biosci. 20.

191-211.

Fugl-Meyer, A. R., Sjostrom, M. and Wahlby, L. (1979) Human plantar flexion strength and structure. Acta physiol. sound. 107, 47-56. Gottlieb. G. L. and Agarwal, G. C. (1971) Dynamic relationship between isometric muscle tension and the electromyogram in man. J. appl. Physiol. 30, 345-351. Hatze, H. (1977) A myocybernetic control model of skeletal muscle. Uiol. C+rncr. 25. 103.~119. Hill, A. V. (1938) The heat of shortening and the dynamic constants of muscle. Proc.. R. Sot. Bl26, 136-195. Hubbard, M. and Trinkle, J. C. (1982) Optimal i,litial conditions for the eastern roll high jump. Eiomeclrunics: Principles und Applic-urions (Edited by Huiskes, R.. Van Campen, D. and De Wijn, J.), pp. 169-174. Martinus Nijhoff, The Hague. Jewell. B. R. and Wilkie, D. R. (1958) An analysis of the mechanical components in frog’s straited muscle. J. Physiol.. Lond. 143, 5 1S-540. Jewell, B. R. and Wilkie, D. R. (1960) The mechanical properties of relaxing muscle. J. Plrysiol., Lund. 152,30-47. Lehman. S. L. (1982) A detailed bioohvsical model of human extraocular ‘muscle. Ph.D. disser;ation, University of California, Berkeley. Lehman, S. L. and Stark, L. (1980) Simulation of linear and non-linear eye movement models: sensitivity analysis and enumeration studies of time optimal control. Cybrrnet. /n/o. Sci. 4, 21-43. Levine, W. S., Zajac. F. E., Belzer. M. R. and Zomlefer, M. R. (1983)Anklecontrols that producea maximal verticaljump when other joints are locked. IEEE Trans. Auromaric Confrol AC 28, 1008-1016. Levine, W. S., Zajac, F. E., Zomlefer. M. R. and Belzer, M. R. (1979) Some experimental, analytical and computational results concerning maximal vertical jumps. Proceedings of 111r17111Annuol A&won Confirmw pp. 904-913. Matsumoto, Y. (1967a) Validity of the force-velocity relation for muscle contraction in the length region, I Q I,. J. gm. PhJsiol. 50, I 125-l 137. Matsumoto, Y. (1967b) Theoretical series elas;ic element length in runa pipiens sartorius muscles.J. gm. Physiol. 50, 1139-1’156. Perrine, J. J. and Edgerton V. R. (1978) Muscle force-velocity and power-velocity relationships under isokinetic loading. Med.

Sci. Sporfs

IO,

159-166.

Ritchie, J. M. and Wilkie. D. R. (1958) The dynamics of muscular contraction. J. PhJsiol., Land. 143, 104-I 13.

Sale. D., Quinlan. J.. Marsh. E.. McComas. A. J. and Belanger, A. Y. (1982) Influence of joint position on ankle plantarflexion in humans. J. appl. Phrsiol. 52. 1636-1642. Taylor, C. P. S. (1969) Isometric muscle contraction and the active state: An analog computer study. Biophys. J. 9, 759-780. Wilkie, D. R. (1956) The mechanical properties of muscle. Br. med. Bull. 12. 177-182. Zajac. F. E., Zomlefer. M. R. and Levine. W. S. (1980) Hindlimb muscular activity. kinetics and kinematics ofcats jumping to their maximum achievable heights. J. rsp. Biol. 91, 73-86.

APPENDIX

A. DERIVATION OF FORWARD JUXIP PERFORIlASCE CRITERIOB

The derivation of the performance criterion for an optimal forward jump is presented here. Struregy and assumptions Let us make the following definitions wt = u,,

“2 = ti,.

(A))

r< = lift-OR time tJ = heel contact time (landing). (1) In order to calculate the performance. J, for a given jump, we will assume that following IiftoBat I,. ankle motion during the airborne phase will not a&et the angular momentum of the upper body significantly. since I, -4 II, (2) Assume that the heel hits the ground in an inelastic collision, and that 0, (I~) = U,. (3) Jumps for which the jumper falls backward after t5 will be disqualified. With the above assumptions. knowledge of the stateat t, is sufhcient to determine the state at I~, and, therefore. the performance or distance jumped. The total state is represcnted by [O,, wt. u,. wz. x, .t, y. J+]*

(AZ)

where x and p are the horizontal and vertical coordinates of the center of mass of the entire body. The position of the toes at I, is chosen to be the origin. Let

a,44 +-

42

(4 +m,)

(A3)

m21,

and “‘8(m1

x,4_x(r,) = -a,

f m,)

cosu,,+a~cosU~,

); = a, sin U,, + a2 sin U,, & = u,wlc

sin U,, - az6.ilr sin Ul,

je = alwlccosUlr

+a2~LccosU2r.

(A4) (AS) (A61

(A7)

In order to determine the horizontal distance travelled during the airborne phase, we need to know the duration of this phase. This is determined by the vertical velocity at time I, and the vertical positions at I, and rd. Let

T, = tJ - I, = duration of airborne phase

then J(t) = y