.I Bwmzchann

co21 9290191 53.00+.00

Vol. 24. No. 1. PP I IO. 1991

C 1991 Pergamon Press

Pruned ,n Grear Brrtam

OPTIMAL

MUSCULAR

COORDINATION JUMPING

STRATEGIES

plc

FOR

MARCUS G. PANDY*~ and FELIX E. ZAJAC$ * Design Division, Mechanical Engineering Department, Stanford University, Stanford, CA 94305-4021, U.S.A.; 1 Rehabilitation Research and Development Center (153), Veterans’ Administration Medical Center, Palo Alto, CA 94304-1200, U.S.A. Abstract-This paper presents a detailed analysis of an optima1 control solution to a maximum height squat jump, based upon how muscles accelerate and contribute power to the body segments during the ground contact phase of jumping. Quantitative comparisons of model and experimental results expose a proximalto-distal sequence of muscle activation (i.e. from hip to knee to ankle). We found that the contribution of muscles dominates both the angular acceleration and the instantaneous power of the segments. However, the contributions of gravity and segmental motion are insignificant, except the latter become important

during the final 10 % of the jump. Vasti and gluteus maximus muscles are the major energy producers of the lower extremity. These muscles are the prime movers of the lower extremity because they dominate the angular acceleration of the hip toward extension and the instantaneous power of the trunk. In contrast, the ankle plantarflexors (soleus, gastrocnemius, and the other plantarllexors) dominate the total energy of the thigh, though these muscles also contribute appreciably to trunk power during the final 20 % of the jump. Therefore, the contribution of these muscles to overall jumping performance cannot be neglected. We found that the biarticular gastrocnemius increases jump height (i.e. the net vertical displacement of the center of mass of the body from standing) by as much as 25 %. However, this increase is not due to any unique biarticular action (e.g. proximal-to-distal power transfer from the knee to the ankle), since jumping performance is similar when gastrocnemius is replaced with a uniarticular ankle plantarflexor.

INTRODUCTION

By analysis of experimental data, several investigators have attempted to gain insight into intermuscular coordination during vertical human jumping (Gregoire et al., 1984; Fukashiro and Komi, 1987; van Ingen Schenau et al., 1987; Bobbert and van Ingen Schenau, 1988). Because non-invasive techniques cannot estimate muscle force accurately, previous efforts have focused on manipulating experimentally measurable quantities (i.e. limb-segmental motions, ground reaction forces, and EMG data) to infer action of muscles on the skeletal system. A complementary approach involves augmenting experimental results with predictions from a model of the same task. In this event, the experimental findings can be used not only to guide model development, but also to gain confidence in interpretations derived from the predictions of the model. Recent work by Bobbert and van Ingen Schenau (1988) and Bobbert et al. (1986a, b) are typical examples of how experimental data are used to infer action of muscles in a motor task. A major hypothesis forwarded by these studies is that jumping performance depends heavily upon the action of biarticular muscles. Specifically, proximal-to-distal power transfer is believed to effect an efficient conversion of rotational motion into translation of the center of Received in final form 28 February 1990. t Present address: Department of Kinesiology and Health

Education, The University of Texas at Austin, Austin, TX 78712, U.S.A.

mass of the body (Gregoire et al., 1984; van Ingen Schenau et al., 1987). The mechanism by which this is accomplished is purported to be the timely activation of rectus femoris (RF) and gastrocnemius (GAS) just prior to lift-off. Activation of RF supposedly decreases the angular acceleration of the trunk, which, combined with the onset of knee extension, results in power being transferred from the hip to the knee (Bobbert and van Ingen Schenau, 1988). Similarly, activation of GAS near lift-off is said to transfer power generated by the knee extensors (e.g. vasti) to the ankle joint (Bobbert and van Ingen Schenau, 1988). Consequently, Bobbert and van Ingen Schenau (1988) claim that the human musculoskeletal system would be equipped with larger (and therefore heavier) knee extensors and ankle plantarflexors in the absence of biarticular muscles. Previously (Pandy et al., 1990), we presented a dynamical model of the human musculoskeletal system and used it to compute the optimal controls, body-segmental motions, muscle forces, and muscle activations for a maximum-height squat jump. By analysing this optimal control solution, the present paper aims to provide a better understanding of how intermuscular control and musculotendon dynamics coordinate jumping. Firstly, we compare the response of the model with experimental data recorded from several subjects performing a maximum-height squat jump. After confirming the ability of the model to accurately reproduce the major features of a maximum-height squat jump, we then analyse in detail the optimal control solution. Specifically, our objectives are to quantify the role of both uni- and biarticular

2

M. G.

PANDY

musculotendon actuators during jumping, focusing upon the contribution of muscles to the vertical acceleration and instantaneous power of each segment during the ground contact phase. An understanding of how muscles contribute to power of a segment is important because the integral of power over time determines the kinetic energy of the segment at lift-off, which affects jump height (see equation (7) in Pandy et al., 1990). METHODS Experiments

with human subjects

Five normal adult males (age 22+2 yr, height 185 + 3 cm, and body mass 78 f4 kg) were chosen as subjects for these experiments. From an initial squatting position, and with hands on hips, each subject performed a two-legged squat jump under the command ‘jump as high as possible’. For three consecutive jumps, force-plate, limb motion, and electromyographic (EMG) data were recorded simultaneously. Ground reaction forces were measured using an AMTI six-component, strain-gaged force platform, having a first natural frequency of 500 Hz (vertical channel). Horizontal and vertical reactions were sampled at 1000 Hz, as were analog EMG data. Pairs of Ag-AgCl surface electrodes (center-to-center distance 3 cm; circumference 12 mm) were attached to the right lower extremity of each subject to record activity in seven muscle groups: soleus (SOL), gastrocnemius (GAS), tibialis anterior (TA), vasti (VAS), hamstrings (HAMS), rectus femoris (RF), and gluteus maximus (GMAX) [see Fig. 1 in Pandy et al. (1990)]. The electrical signals were amplified (Grass Instruments Inc., type P51 lK), and then band-pass filtered (cutoff frequencies 50 Hz and 400 Hz) to minimize errors introduced by noise and motion artifact. To record the limb-segmental angular displacements of each subject, seven light-emitting diodes (LEDs) were positioned over six bony prominences: fifth metatarsophalangeal joint, calcaneus, lateral malleolus, lateral epicondyle, greater trochanter, and glenohumeral joint. Together these landmarks defined four body segments: the foot, shank, thigh, and HAT segment (head, arms, and trunk). Using the Selcom Selspot II active-marker system, absolute displacements were recorded at 312 Hz (i.e. displacements of each landmark were referenced to a global frame attached to the force platform). All post-processing of the data was carried out on a PDP-11/34 digital computer. Contribution of muscles to the acceleration and power of segments

The dynamical equations of motion can be used to find the contribution of a muscle to the acceleration of the segments and mass centroid (Gordon et al., 1986; Zajac et al., 1986; Hatze, 1987; Zajac, 1987; Zajac and Gordon, 1989). The equations can also be used to

and F. E. ZAJAC

calculate the contribution of muscle to the instantaneous power of segments (e.g. Aleshinsky, 1986). Appropriate to jumping, and using the dynamical equations of motion for the four-segment, planar, skeletal model, the vector of limb-segmental angular accelerations is (Pandy et al., 1990): &=/t(B)-‘{B(0)8*+C(B)+DM(B)Pr+T(B,8)}.

(1)

Each element of the vector DM(B)Pr is the sum of the segmental torques developed by the muscles attaching to or crossing the corresponding body segment. The total contribution of muscles to the vector of segmental accelerations #,,, is thus: &,,=A@)-‘aDM(B)Pr.

(2)

Since A(8)- ’ is, in general, non-diagonal, notice that each muscle force accelerates all body segments. Thus, muscles not directly attached to or spanning a segment still contribute to its acceleration (see Zajac and Gordon (1989) for review). Knowing the total instantaneous power of a segment (&= dE/dt, which is the time rate of change of the total mechanical energy of a segment) (Meriam, 1975), we computed the contribution of muscles, gravity, and segmental motion (‘inertia’) to the instantaneous segmental power. For example, the contributions to trunk power are (see Appendix): Bi,,,,i,=mu(e)A(e)-1{B(e)82+mu(8,8)} -v4 &svity =

Cmgj

&l”sclc

[mu(e)A(e)-1DM(e)PT].~4

=

+ m4cw)-

+ FEW’ ~‘I&I + FGMAX

’ cm

+ FHAM

’ bGMAX)rel.

-v4

’ h~drc~

(3)

Note that Eincrtir contains the contributions from the centrifugal and Coriolis forces due to the motion of the body [i.e. the vectors B(8)8* and mu], whereas _$lavi,yresults from the gravitational force. &,usc,e, on the other hand, is the net contribution of muscles to the instantaneous power of the trunk. The net contribution of muscles is composed of contributions from those intersegmental forces acting at the hip joint that arise from muscle forces, as well as from ‘direct’ muscular forces that arise from the insertion of muscle on that segment [e.g. F,, - (v,&,] (Aleshinsky, 1986). Since the matrix A(B)- ’ appears throughout equation (3), this means that muscles not directly attached to the trunk still contribute to its power. This result follows directly from equation (l), where A(8)-’ couples gravity, inertia forces, and muscle forces into all body-segmental motions.

RESULTS

Comparison of model and experiment

The foot, shank, thigh, and trunk angles of the model are plotted in Fig. 1 (heavy solid lines), with the shaded areas representing the range of the subjects’

Muscular coordination of jumping force (% budy 3w weight)

displacement

(Ml

I ”

20

3

$0

5%

80

100

% of ground contact time Fig. 1. Model (heavy solid lines) and experimental (shaded areas and light solid lines) limb-segmental angular displacements during the ground contact phase ofjumping (O-100 %). The shaded areas represent the range of experimental data for five subjects performing a maximum-height squat jump, whereas the light solid lines are the computed means of these data. For each subject, 0 % of ground contact time defines the instant that the vertical ground reaction force decreases to 95 % of body weight. For the model, prior to and at 0 % ground contact time, muscle forces are constant to maintain the body statically in the squat. All angles are defined in Fig. 2 of Pandy et al. (1990).

experimentally recorded limb-segmental angular displacements. The means of these experimental data have also been calculated and plotted in Fig. 1 (light

solid lines). The measured (shaded regions and light solid lines) and predicted (heavy solid lines) vertical and horizontal ground reactions are given in Fig. 2. Once again, the shaded regions actually represent the range of our experimental data, while the light solid lines are computed means of these data. Note that all time scales have been normalized to compare one jump to another. In general, there is good agreement between model and experiment. The average jump height and lift-off time for the five subjects were 35+ 5 cm and 0.60 +O.lO s respectively. Corresponding values for the model are 33 cm and 0.50 s. For each subject, 0 % of ground contact time was taken to be the instant the vertical ground reaction decreased to 95 % of body weight. Those subjects who jumped higher than the

Fig. 2. Model (heavy solid lines) and experimental (shaded areas and light solid lines) ground reaction forces during the ground contact phase. The shaded areas represent the range of ground reaction forces generated by five subjects performing a maximum-height squat jump. The light solid lines are the computed means. Positive force indicates that the body is pushing downwards (top figure), or backwards (bottom figure). No significance should be attached to the two peaks at 75 and 95 % of ground contact time in the horizontal ground reaction force of the model (heavy solid line).

model did so because of their ability to apply a vertical force to the ground for a longer period of time (thereby imparting a larger vertical impulse to the center of mass of the body). The ground reactions predicted by the model (heavy solid lines in Fig. 2) are also in good agreement with those generated by our subjects. In both cases, peak vertical forces are approximately 2.2 x body weight. Both show the existence of a countermovement prior to upward propulsion, which occurs during the first 40 % of jump time (Fig. 2, vertical force; compare thin solid line with heavy solid line). Since the subjects were not told to jump ‘straight up’, but rather to ‘jump as high as possible’, it appears that the subjects chose to countermove their bodies because that causes them to jump higher, just as the model predicts. In addition, both the model and the subjects generate a biphasic horizontal ground reaction force (Fig. 2, horizontal force). The latter agreement is particularly encouraging since the horizontal ground force, by being only 30 % of body weight, is presumably sensitive to the way the jump is executed. There are, however, differences in the rates at which the ground reaction forces are applied (compare the light and heavy solid lines near lift-off). Specifically,

4

M. G. PANDYand F. E. ZAJAC

OPF

-I

7-A I GAS

I

I

RF

I

HAMS

SOL

GMAX I

0

20

40 %

M

en

,lll

of groundconroct time

Fig. 3. Experimental EMG activity (light wavy solid lines) from one subject and neuromuscular control signals predicted by the model (heavy solid lines) during the ground contact phase. Notice that muscles are recruited, in general, proximally to distally. No EMG activity was recorded from OPF because they are deep muscles of the calf. No significance should be given to the occassional spikes in the control signals (e.g. RF), which arise because of computational imprecision. The light straight lines indicate the zero level for both the EMG activity and the controls.

the model predicts a steeper increase in the vertical reaction during initial propulsion, together with a more rapid decrease prior to lift-off. Because the angular velocities of the limb segments in the model increase just prior to lift-off (Pandy et al., 1990), the vertical ground reaction decreases very rapidly in an attempt to maximize the impulse applied to the mass centroid. In reality, joint angular velocities decrease near lift-off (Bobbert and van Ingen Schenau, 1988), and thus the measured vertical reaction forces decrease sooner than predicted by our model. Experimental EMG activity was found to agree qualitatively well with the computed optimal controls (e.g. Fig. 3). However, minor differences were found. For example, the optimal solution predicts GMAX to begin excitation at 30 % of ground contact time and to remain fully active until lift-off (heavy solid line in Fig. 3, GMAX). By comparison, recorded EMG data indicate very low activity of GMAX prior to lift-off (light wavy solid line in Fig. 3, GMAX), presumably to prevent hip hyperextension after lift-off (Bobbert and

van Ingen Schenau, 1988). Further, our EMG results suggest that HAMS is activated at lift-off, whereas the optimal control solution predicts inactivity at this time (Fig. 3, HAMS). Though the foot, shank, and thigh angles of the model compare well with measured data (Fig. l(a), (b), and (c) show these results to be within S’), a relatively large difference exists at the trunk [compare heavy and light solid lines, Fig. l(d)]. The trunk of the model undergoes very little countermovement [less than Y, Fig. l(d)] prior to upward propulsion, whereas all our subjects countermove their upper bodies by as much as 10”. One reason why the trunk of the model moves little may be due to the one-segment representation for the trunk. Such a representation may indeed be an unrealistic simplification of body-segmental dynamics [e.g. see Hatze (1981a)], especially if spinal flexure is significantly utilized during upward propulsion. Taking this feature into account will require two or more segments to model the trunk, together with detailed representations of both upper- and lowerback musculature. Such changes will demand an appreciable increase in overall model complexity. A second explanation for the absence of trunk countermovement in the model involves the parameters chosen to describe the musculotendon dynamics for the actuators spanning the hip (particularly GMAX). Specifically, countermovement may not be tolerated because of the inability of the model to generate sufficient hip moment. With the aim of strengthening the hip musculature of the model, we tried increasing the peak isometric strength of GMAX by 20 %. This increases muscle force (and therefore moment) generated in the range of hip-joint motion experienced during jumping. Unfortunately, increased hip strength has failed to produce the desired increase in countermovement of the trunk in the simulation. In our search to replicate trunk countermovement, we are currently altering the tendon slack lengths of hip muscles, which shift the moment-angle curves (Hatze, 1981b; Hoy et al., 1990). We are also increasing tendon compliance of hip muscles, though countermovement is unlikely to be found since hip muscles are stiff, in an absolute sense, and in a relative sense as well, compared to the muscles spanning either the ankle or knee (Zajac, 1989). Intermuscular

coordination

Since the response of the model replicates the major features of the jump, it can be analysed to understand coordination. We specifically aim to quantify the contribution of muscles to the vertical acceleration and the instantaneous power of the trunk during the ground contact phase. The reason for focusing on the trunk is that it represents approximately 70 % of total body mass. Though the results discussed above indicate that the model is imperfect in replicating the exact response of the trunk [Fig. l(d)], the model does indeed generate levels and trajectories of horizontal

Muscular

trunk

coordination

acceleration

20

% of ground

contact

time

Fig. 4. Contributions of all the muscles (‘muscle’), gravity (‘gravity’), and segmental motion (‘inertia’) to the total angular acceleration of the trunk (shaded area) computed from the model during the ground contact phase. Notice that muscles dominate, except near lift-off. Positive angular acceleration is a backward rotation of the trunk.

segment

5

of jumping

and vertical acceleration of the mass centroid of the body that are consistent with experimental observations, as evident by the horizontal and vertical ground reaction forces, respectively (Fig. 2). Thus, we feel that the model can be used to understand how muscles accelerate and transfer power to the trunk. Muscles were found to dominate the angular acceleration of the trunk for all but the final 10 % of the jump (compare heavy solid line with shaded area, Fig. 4). In the final 10 % of ground contact time, the contribution of the segmental motions to the acceleration of the trunk outweigh those of muscles since segmental angular velocities increase rapidly near liftoff. Similar results were found for the foot, shank, and thigh. Because muscles dominate the angular accelerations of the segments for almost the entire jump, analysing individual muscular contributions (Figs 5 and 6) allows us to better understand muscle-force sequencing during jumping. Notice that TA, HAMS, and RF make negligible contributions to the joint angular accelerations (dashed lines, Fig. 6). The muscles that dominate the joint angular accelerations are GAS,

acceleration (radlse6) joint

acceleration

zm

I 0

20

40

% of ground

80

60

contact

100

time 20

Fig. 5. Contributions of muscles to the angular acceleration of the trunk, thigh, and foot derived from the model during the ground contact phase. The shaded region is the summed effect by all the muscles. UPF is the combined contribution of the uniarticular ankle plantarflexors (SOL and OPF). Light sohd lines in (a) are used only for clarity. Positive angular acceleration is a backward rotation of the trunk (a] and a forward rotation of the thigh (b) and foot (c).

40

% of ground

60

contact

80

IO0

time

Fig 6. Contributions of muscles to the angular acceleration of the hip, knee, and ankle derived from the model during the ground contact phase. The shaded region is the summed effect by all the muscles. The dashed lines represent the combined effect of TA, HAMS, and RF. Positive angular acceleration is joint extension.

6

M. G. PANDY and F. E. ZAJAC

VAS, GMAX, and the uniarticular ankle plantarflexors SOL and OPF (designated by UPF). Of all these muscles, only one (i.e. GAS) is biarticular. HAMS and GMAX accelerate the trunk towards an upright position (i.e. positive angular acceleration) to counteract gravity [Fig. S(a)], whereas VAS, SOL, OPF, and GAS induce opposing angular accelerations. Activation of HAMS and GMAX (the hip extensors in our model) are therefore crucial to counteracting the effects of gravity on the trunk during the initial phase of propulsion (i.e. the first 60 % of the jump). Thus, HAMS and GMAX are excited early (about 20-30 % of ground contact time, Fig. 3 HAMS and GMAX) to initiate desirable backward (positive) rotation of the trunk [Fig. 5 (a)]. VAS is excited later (about 40 % into the jump, Fig. 3 VAS) because it can only induce undesirable forward (negative) angular acceleration of the trunk prior to 80 % of ground contact time (Fig. 5 (a) VAS). Later (8CrlOO % of ground contact time), positive acceleration from VAS is possible because the body segments are closer to colinearity. VAS can then assist GMAX in positive rotation of the trunk [Fig. 5(a)]. Even though VAS induces a negative angular acceleration of the trunk during 80 % of ground contact time, this knee extensor actually accelerates the hip towards extension (Fig. 6(a) VAS) because of the large positive angular acceleration of the thigh it induces (Fig. 5(b) VAS). Since VAS is assumed to be the strongest muscle in the lower extremity [see Table 1 in Pandy et al. (1990)], any action other than hip extension would clearly be undesirable. Thus, GM AX and VAS, the strongest hip and knee extensors in the model [see Table 1 in Pandy et al. (1990)] work together to produce angular acceleration of the knee and hip towards extension [Fig. 6(a)], which acts to accelerate the body mass in the desired direction (upwards).

Unfortunately, the above actions simultaneously accelerate the heel into the ground [see contributions of VAS and GMAX to negative angular acceleration of the foot, Fig. 5 (c)]. In response, SOL and OPF [i.e. UPF in Fig. 5 (c)] are excited at 50-60 % of ground contact time (Fig. 3 SOL and OPF). SOL and OPF also counter the large ankle dorsiflexor accelerations induced by VAS and GMAX [Fig. 6(c)]. Finally, GAS is excited near lift-off (Fig. 3 GAS) to further accelerate the ankle towards plantarflexion [Fig. 6(c)], which arises from the positive acceleration of the foot it induces [Fig. 5 (c)l. In so doing, GAS (and the UPF as well) opposes the efforts of VAS and GMAX at this time [e.g. GAS accelerates the knee towards flexion; Fig. 6(b)]. Interestingly, the increase in performance resulting from increasing foot angular velocity near lift-off outweighs the undesirable effects (i.e. flexion of the knee and hip rather than extension). That is, the net effect of GAS is to increase the vertical acceleration of the mass centroid of the body, and thus GAS acts to increase the vertical velocity of the body at lift-off. Role of uni- and biarticular muscles

VAS and GMAX were found to be the major energy producers, the prime movers, of the lower extremity during jumping, since they contribute most significantly to the total energy made available for propulsion (Fig. 7, areas under curves). Note that, in the final 20 percent, the ankle plantarflexors UPF and GAS contribute significantly as well. A large proportion of the total energy delivered to the segments (Fig. 8, shaded region) resides in the trunk (Fig. 8, compare the areas under the individual curves to the shaded region). In fact, the combined energy of the thigh, shank, and foot amounts to only 30 % of the total energy available at lift-off (with approximately half of this belonging to the thigh). Therefore, a large portion (i.e. 70 %) of the input musculotendon energy is transferred to the trunk. This is not a surprising result, given that the trunk represents almost 70 % of the body mass.

instantaneous segmental power (watts) 3lXM I

JYk--e-

m

40

\ 60

80

Ia3

90 of ground contact time Fig. 7. Mechanical power generated by the musculotendon actuators included in the model during the ground contact phase. Power was computed by multiplying musculotendon velocity with tendon force. UPF represents the combined power of the uniarticular ankle plantarflexors (SOL and OPF). Light solid lines are used only for clarity. Energy supplied by an actuator is the area under the curve. Notice that GMAX and VAS are the major energy producers.

0

20

40

64

80

IW

90 of ground contact time Fig. 8. Instantaneous power of each segment in the model during the ground contact phase. The shaded area is the summed power of all four segments. The area under each curve is the energy of that segment at body lift-off. Notice that most of the energy resides in the trunk at lift-off.

Muscular coordination of jumping

7

then (Pandy et al., 1990). The contribution of gravity is insignificant throughout propulsion. Of all the muscles, VAS and GMAX contribute

trunkpowerfrom 2lnm

thigh

muscles

ower

fkltts)

I 0

20

60

60

80

ID0

% of ground contact time

Fig. 9. Muscular, gravitational, and inertial contributions to the power of the trunk (a), and individual muscular contributions to trunk power (b) and to thigh power (c). (a) Total power of the trunk is the shaded curve. ‘Muscle’ is the contribution from the muscles, ‘gravity’ from gravity, and ‘inertial’ from the motion of all the segments. Notice that muscles dominate. (b) Total contribution from all the muscles is the shaded curve. PF is the contribution of the ankle plantarflexors (SOL, OPF, and GAS). The dashed line is the combined contribution of HAMS, RF, and TA. The area under the curve is the energy developedby each muscle (or muscle group) that is transferred to the trunk. Notice that VAS and GMAX are the major contributors. (c) Total contribution from all the muscles is the shaded curve. PF is the contribution of the ankle plantarflexors (SOL, OPF, and GAS). The dashed line is the contribution from all other muscles. Notice that the ankle plantarllexors are the major contributors to thigh power.

Muscles dominate the contribution to the instantaneous power of the trunk (Fig. 9(a), compare heavy solid curve to the shaded region). However, the contribution of segmental motion approximates the contribution of muscles near lift-off (compare ‘muscle’ curve to ‘inertial’ curve at 100 %) because the angular velocities of the segments of the model are quite large

most significantly to the instantaneous power of the trunk [Fig. 9(b)], and account for approximately 90% of the total energy of the trunk at lift-off [summed area under the curves for VAS and GMAX in Fig. 9 (b) compared to the area in the shaded region in Fig. 9 (a)]. Moreover, almost all of the total energy generated by the prime movers VAS and GMAX is made available to the trunk at lift-off [compare the areas under the curves for VAS and GMAX in Fig. 7 and Fig. 9(b)]. The ankle plantarflexors (SOL, OPF, and GAS) make their presence felt only during the final 20 % of the jump [the PF curve in Fig. 9(b)] by contributing about 30 % to the total power delivered by muscles to the trunk (Fig. 9(b), compare the PF curve with the shaded region at lift-off). Note that the ‘hump’ in the trajectory of total power (Fig. 9 (b), 90 % of ground contact time), is due primarily to the contribution of the ankle plantarflexors. In addition, the ankle plantarflexors, by dominating thigh power, except near liftoff, contribute approximately 90 % to the peak power of the thigh, which occurs at about 85 % of ground contact time [Fig. 9(c)]. Because of the combined effects of the ankle plantarflexors on the trunk near lift-off and on the thigh at other times, we conclude that the contribution of the ankle plantarflexors to overall jumping performance cannot be neglected. Of the ankle plantarflexors, GAS is an important contributor. For instance, it contributes about the same amount of power as SOL and OPF (i.e. approximately 30 %) to each of the thigh and trunk [Fig. 9 (b)-(c)]. It should be mentioned that GAS actually delivers power to the thigh, even though it accelerates the knee towards flexion [Fig. 6(b)], because of the substantial power being delivered to the thigh through the intersegmental forces acting at the knee joint. To conclude, of the biarticular muscles, GAS is expected to contribute somewhat to overall jumping performance. Finally, to assess the dependence of jumping performance on biarticular muscle function, we removed all biarticular muscles from the model [see Fig. 1, Pandy et al. (1990)] and, with only uniarticular muscles available for propulsion, re-computed how muscles would then have to be coordinated to produce as high a squat jump as possible. We found that even though the jump was still coordinated (Fig. 10, stick figures), jump height decreased [compare the solid and dashed lines in Fig. 10 at lift-off (100 %)I. In fact, the net vertical displacement of the mass centroid of the body from standing decreases by 20 %, supporting our previous conclusion that the contribution of the biarticular muscle GAS to jumping performance cannot be neglected. However, we do not feel that the mechanism by which GAS contributes to the jump is any different from that of uniarticular muscles (see below).

8

M. G.

PANDY

pofenrial ‘ump

$=I:;:-----0

20 % 0fgrLd

conTacttimr

100

Fig. 10. Simulated jump using only uniarticular muscles. HAMS, RF, and GAS were removed from the model and, given this new set of muscles, another simulated jump was found using the optimization algorithm. The stick figures show how the body segments should be coordinated during the ground contact phase (O-100 %) to achieve the highest possible height with the new muscle set. The heavy solid and dashed lines show how high the mass centroid of the body would go if the body could leave the ground at that time, given that all the muscles, or just uniarticular muscles, are used to coordinate the jump, respectively. The thin dashed line is the standing height of the simulated jumper.

DISCUSSION

Though the use of optimal control theory to elucidate the control of the musculoskeletal system is widely appreciated (Hatze, 1976; Davy and Audu, 1987), its full potential has yet to be realized. The major reason is that the human musculoskeletal system has high dimensionality, and exhibits severe nonlinearities. Consequently, even moderately complex optimal control problems encounter serious computational difficulties. We have nevertheless formulated and solved an optimal control problem for maximum-height jumping (Pandy et al., 1990) in the hope of understanding how intermuscular control and musculotendon dynamics coordinate complex human movements. By conducting a detailed analysis of the optimal control solution, we have been able to investigate intermuscular coordination of the squat jump, knowing the time histories of all muscle forces, muscle activation levels, and body-segmental motions. This represents a most important advantage over previous efforts to analyse coordination of jumping, since those efforts focused on relatively small portions of the human musculoskeletal system (Bobbert et al., 1986a, b). Our optimal control solution has predicted a proximal-to-distal sequence of muscle activation, where muscles spanning the hip (i.e. GMAX and HAMS) are activated first, followed by uniarticular actuators spanning the knee (i.e. VAS) and ankle (i.e. SOL, OPF). This trend is consistent with our experimental

and F. E.

ZAJAC

findings (Fig. 3), as well as with previous experimental findings (Gregoire et al., 1984; Bobbert and van Ingen Schenau, 1988). Gregoire et al. (1984) and Bobbert and van Ingen Schenau (1988) conclude, however, that a proximalto-distal sequence of movements allows an energy flow from the hip to the ankle joint. Gregoire et nl. (1984) have stated that uniarticular muscles (e.g. GMAX) generate power at high contraction velocities, and that this power is transferred by biarticular muscles to distal joints at low contraction velocities (e.g. transfer of power by RF from the hip to the knee). In fact, Bobbert et al. (1986b) have suggested that two factors contribute to the large ankle power output observed during one-legged jumping: compliant tendons and power transfer via biarticular muscles. They found that elastic recoil in tendinous structures account for up to 40 % of total ankle power, whereas power transfer from the knee to the ankle by GAS contributes approximately 25 %. Consequently, Bobbert et al. (1986b) have concluded that the biarticular action of GAS increases jumping performance significantly. Moreover, these findings have been punctuated by the hypothesis that jump height would decrease by as much as 50 % if only uniarticular muscles were available for propulsion (Gregoire et al., 1984). Our results are in opposition to the notion that energy flows distally. For example, we found that energy production by musculotendon actuators is dominated by the uniarticular extensors VAS and GMAX (Fig. 7), and that almost all of this energy is used to accelerate the center of mass of the trunk upwards [Fig. 9(b)]. In this respect, our interpretation of the role of these uniarticular muscles differs from that of previous investigators (Gregoire et al., 1984; Bobbert and van Ingen Schenau, 1988). We have found no evidence to support the contention that power generated by the prime movers (i.e. VAS and GMAX) is transferred distally to the ankle joint. To the contrary, the major role of these muscles is to deliver power proximally to the massive trunk segment in an effort to maximize the vertical velocity of the mass centroid of the body at lift-off. In fact, our results have shown that the ankle plantarflexors (SOL, OPF and GAS) also deliver power to proximally located segments (i.e. the trunk and thigh), further supporting the claim that lower extremity muscles act to transfer power to the most massive body segments during upward propulsion. Of all the lower extremity biarticular muscles, we believe that GAS is the most important. Figure 10 predicts a 20% decrease in performance when only uniarticular muscles are used for propulsion. Even though the energy produced by RF and HAMS are not shown in Fig. 9(b)-(c), our data show that the biarticular RF and HAMS contribute less than GAS to the total energy of the trunk and thigh. In fact, GAS contributes 20 % to the total thigh energy at lift-off, whereas RF and HAMS contribute less than 10%.

9

Muscular coordination of jumping

Moreover, while GAS supplies only a small fraction of the total energy of the trunk at lift-off (i.e. less than 5 W), RF and HAMS combine to remove energy from this segment. We claim, therefore, that the absence of GAS is the major factor contributing to the decrease in jump height shown in Fig. 10. In this regard, our results support a previous conclusion (Bobbert et al., 1986b; Bobbert and van Ingen Schenau, 1988) that the contribution of GAS to jumping performance cannot be neglected. We are, however, opposed to the notion that jumping performance is increased by the unique biarticular action of GAS. Our results do not substantiate the claim that ankle power output is increased significantly as a result of power transferred by GAS from the knee to the ankle. In fact, by delivering power to the trunk and thigh, GAS behaves no differently from any of the uniarticular ankle plantarflexors (SOL and OPF). So, while it is true that GAS acts to increase jumping performance appreciably (i.e. by as much as 25 %), this increase is not a consequence of GAS being biarticular (i.e. the increase is not due to proximal-todistal power transfer from the knee to the ankle). To support this claim, we offer the following evidence. Upon finding a decrease in performance when only uniarticular muscles are used for propulsion (Fig. lo), we added a uniarticular ankle plantarflexor that was structurally identical to GAS. With the biarticular capability of GAS to transfer power now removed, we found jumping performance to be restored (and in fact increased). We conclude, therefore, that GAS is activated near lift-off to increase foot angular velocity (and ultimately the vertical velocity of the mass centroid of the body). More importantly, the action of GAS in this task is no different from that of any other uniarticular extensor muscle; it too transfers power in a proximal direction to the trunk and thigh. Acknowledgements-We gratefully acknowledge discussions we have had with William S. Levine on jumping and optimal control theory, and thank him for a careful review of an earlier version of this manuscript. We are also indebted to Melissa Hoy, Scott Tashman, and Pamela Stevenson for their contributions to the jumping experiments, Arthur Kuo for his help in analysing the computer results, and David Delp for figure preparation. This-work was supported by NIH grant NS17662, the Alfred P. Sloan Foundation, and the Rehab. R&D Service, Dept. of Veterans’ Affairs.

REFERENCES

Aleshinsky, S. Y. (1986) An energy ‘sources’ and ‘fractions’ approach to the mechanical energy expenditure problem-11. Movement of the multi-link chain model. J. Biomechnnics 19, 295-300. Bobbert, M. F., Huijing, P. A. and Ingen Schenau, G. J. van (I 986a) A model of the human triceps surae muscle-tendon complex applied to jumping. J. Biomechanics 19,887-898. Bobbert, M. F., Huijing, P. A. and Ingen Schenau, G. J. van (1986b) An estimation of power output and work done bv the human triceps surae muscle-tendon complex in jump ing. J. Biomechanics 19, 899-906.

Bobbert, M. F. and Ingen Schenau, G. J. van (1988) Coordination in vertical jumping. J. Biomechanics 21, 249-262. Davy, D. T. and Audu, M. L. (1987) A dynamic optimization technique for predicting muscle forces in the swing phase of gait. J. Biomechanics 20, 187-201. Fukashiro, S. and Komi, P. V. (1987) Joint moment and mechanical power flow of the lower limb during vertical jump. Int. J: Sports Med. 8, 15-21. Gordon. M. E.. Zaiac. F. E. and Hov. M. G. (Soon: Carter. M. C.j (1986) Post&al synergies +&ctated ‘b; segmental accelerations from muscles and physical constraints. Neurosci. Abstr. 12, 1425. Gordon, M. E., Zajac, F. E., Khang, G. and Loan, J. P. (1988) Intersegmental and mass center accelerations induced by lower extremity muscles: theory and methodology with emphasis on quasi-vertical standing postures. In Computational Methods in Bioengineering, I988 ASME Winter A. Mtg, Chicago (Edited by Spilker, R. L. and Simon, B. R.), Vol. 9, pp. 481-492. Gregoire, L., Veeger, H. E., Huijing, P. A. and Ingen Schenau, G. J. van (1984) Role of mono- and biarticular muscles in explosive movements. Int. J. Sports Med. 5, 301-305. Hatze, H. (1976) The complete optimization of a human motion. Mathl Biosci. 28, 99-135. Hatze, H. (1981a) A comprehensive model for human motion simulation and its application to the take-off phase of the long jump. J. Biomechanics 14, 135-142. Hatze, H. (1981b) Estimation of myodynamic parameter values from observations on isometrically contracting muscle groups. Eur. J. appl. Physiol. 46, 325-388. Hatze, H. (1987) Gait analysis: adequacy of current models and research strategies. J. Motor Behaoior 19, 28&287. Hoy, M. G., Gordon, M. E., Zajac, F. E. and Maclean, K. E. (1990) A musculoskeletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle relationship of musculotendon actuators at the hip, knee, and ankle. J. Biomechanics 23, 157-169. Ingen Schenau, G. J. van, Bobbert, M. F. and Rozendal, R. H. (1987) The unique action of biarticular muscles in complex movements. J. Anat. 155, 1-5. Meriam, J. L. (1975) Dynamics, 2nd Edn. Wiley, New York. Pandy, M. G., Zajac, F. E., Sim, E. and Levine, W. S. (1990). An optimal control model for maximum-height human jumping. J. Biomechanics 23, 1185-l 198. Zajac, F. E. (1987) Dynamics of limb movement: interpretation of EMG signals and the effects of musculotendon forces on body acceleration. In ASME 1987 Biomech. Symp., New York (Edited by Butler, D. L. and Torzilli, P. A.), Vol. 84, pp. 391-394. ASME, New York. Zajac, F. E. (1989) Muscle and tendon: properties. models. scaling, and application to biomechanics and motor control. In CRC Crit. Rev. Biomed. Engng (Edited by Bourne, J. R.), Vol. 17, pp. 359411. CRC Press, Boca Raton. Zajac, F. E. and Gordon, M. E. (1989) Determining muscle’s force and action in multi-articular movement. In Exercise and Sport Science Reviews (Edited by Pandolf, K.), Vol. 17, pp. 187-230. Williams and Wilkins, Baltimore. Zajac, F. E., Gordon, M. E. and Hoy, M. G. (1986) Physiological classification of muscles into agonist-antagonist muscle action groups: theory and methodology based on mechanics. Neurosci. Abstr. 12, 1424.

APPENDIX

The following is a derivation of the equations defining the total instantaneous power of the trunk. Consider a free-body _ . diagram of the trunk with the following muscles attached: HAMS, RF, and GMAX (Fig. Al). The total instantaneous

10

M. G.

PANDY

and F. E. ZAJAC functions.of the limb-segmental angular displacements only; and u(0,0) is a (2 x 1) vector containing entries which are functions of both segmental angular displacements and velocities. Substitute equation (1) into equation (A4) to get: R=u(e)A(e)-'{B(e)Bz+c(e)+DM(e)PT+T(e,B)}+v(e,8). (A5) Then, substituting equation (A5) into equation (A3) gives: F, = - Fxr - FHAM- FGMAX+ mgi +mu(e)A(B)-'{B(e)BZ+c(e)+DM(e)PT+T(e,B)j

Fig. Al. Free-body diagram of the trunk segment. All symbols are defined in the text.

+mo(e,d).

power of a segment is given by (Meriam, 1975): ti=ZF*v+Zz*o

(Al)

where F is any external force acting on the segment, v is the linear velocity of the point on the segment at which the force F is applied, T is any external torque acting on the segment, and Q)is the angular velocity of the segment. Using equation (Al), the total instantaneous power of the trunk is: E,=F,*v,+Fs,*

vRF+FHAM'v~A~

+FG,,X*vG,x

- FGMAx+ mgj + ma

(A3)

where j is a unit vector in the vertical direction (positive upwards). Now write the acceleration of the center of mass of the trunk as: n=u(B)B’+v(B,B)

vrW= v4 + (vs,),,,

(A7)

VHAM = v4 + (v,,,),,,

(AS)

vGMAX=v4+(vGP.,.4X)r,,

(A9)

where (v,,),,, is the velocity of the point of attachment of RF relatioe to the hip joint; and similarly for HAMS and GMAX respectively. Finally, substitute equations (A7HA9) into equation (A2) and rearrange to get:

(-42)

where F, is the intersegmental force acting at the hip joint; vq is the linear velocity of the hip joint; F,,, F,,,, FoMAXare forces generated by RF, HAMS, and GMAX respectively; and VRF~~HAMIVGMAX are the velocities of the points of attachment of RF, HAMS, and GMAX on the trunk. Using Newton’s laws, the intersegmental force F, can be found directly from Fig. Al: F4 = - F,, - F,,,

(W

But the velocity of the point of attachment of a muscle in equation (A2) may be expressed in terms of the hip joint velocity:

(A4)

where u(0) is a (2 x 4) matrix containing entries which are

+FGMAX '(VGMAX)rcl+[mgj+mu(e)A(e)-' x {B(e)ez+c(e)+DM(e)PT+T(e,f$} +mv(e,f$].v,.

(AlO)

Equation (AlO) defines the total instantaneous power of the trunk. The inertial, gravitational, and muscular contributions are: Lti.

= mu(e)A(e)-1{8(e)82+mu(e,6j)}.~,

~R,..e,=[mgi+mu(e)A(e)-'c(e)l.v4 = [mu (ey 6n"SCk

(e)- 1md(e)p7 .v4

+FRF'(~RF),,, +FHAM'~A&, +FGMAx'(~GMAX)~~P

(All)

Optimal muscular coordination strategies for jumping.

This paper presents a detailed analysis of an optimal control solution to a maximum height squat jump, based upon how muscles accelerate and contribut...
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