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002, 9290/93 %6.W + .oO T-) 1993 Pergamon Press Ltd J Biomechanics Vol 26. No. 12. pp 1413 1427, 1993 Printed m Great Britain STORAGE AND UTILIZATION OF ELASTIC STRAIN ENERGY DURING JUMPING FRANK *Department C. ANDERSON* and MARCUS G. PANDY*~ of Kinesiology and Health Education, and TDepartment of Mechanical The University of Texas at Austin, Austin, TX 78712, U.S.A. Engineering, Abstract-Based upon the optimal control solutions to a maximum-height countermovement jump (CMJ) and a maximum-height squat jump (SJ). this paper provides a quantitative description of how tendons and the elastic elements of muscle store and deliver energy during vertical jumping. After confirming the ability of the model to replicate the major features of each jump (i.e. muscle activation patterns, body-segmental motions, ground reaction forces, jump height, and total ground contact time), the time histories of the forces and shortening velocities of all the musculotendon actuators in the model were used to calculate the work done on the skeleton by tendons as well as the series-elastic elements. the parallel-elastic elements. and the contractile elements of muscle. We found that all the elastic tissues delivered nearly the same amount of energy to the skeleton during a CMJ and an SJ. The reason is twofold: first, nearly as much elastic strain energy was stored during the SJ as the CMJ; second, more stored elastic strain energy was lost as heat during the CMJ. There was also a difference in the way energy was stored during each jump. During the CMJ. strain energy stored in the elastic tissues came primarily from the gravitational potential energy of the skeleton as the more proximal extensor muscles were stretched during the downward phase of the jump. During the SJ, on the other hand, energy stored in the elastic tissues came primarily from the contractile elements as they did work to stretch the tendons and the series-elastic elements of the muscles, Increasing tendon compliance in the model led to an increase in elastic energy storage and utilization, but it also decreased the amount of energy delivered by the contractile elements to the skeleton. Jump height therefore remained almost the same for both jumps. These results suggest that elastic energy storage and utilization enhance jumping efficiency much more than overall jumping performance. INTRODUCTION If an activated muscle is stretched prior to shortening, its performance is enhanced during the concentric phase. This phenomenon, which has been demonstrated in in vitro and in viuo muscle experiments (Asmussen and Bonde-Petersen, 1974a; Cavagna et al., 1968; Komi and Bosco. 1978), is purported to be the result of strain energy stored in the elastic tissues of muscle. One activity in which elastic energy storage and utilization is thought to be important is vertical Several experiments have shown that jumping. humans typically jump higher when propulsion is preceded by a preparatory countermovement. The hypothesis forwarded to explain this result is that lengthening of the extensor muscles during a countermovement leads to an increase in the amount of energy stored in the elastic tissues, which in turn increases the energy delivered to the skeleton during the propulsion phase of the jump (Asmussen and Bonde-Petersen, 1974a; van Ingen Schenau. 1984; Komi and Bosco, 1978). Caution should be exercised, however, when interpreting the results of jump-height experiments. In particular, it has never been shown that the difference in jump height between countermovement and noncountermovement jumps is due to the utilization of stored elastic strain energy. To examine whether or not utilization of elastic strain energy increases Received in final form BM 26:12-E 17 March 1993. jumping performance and, more importantly, to understand how the elastic tissues enhance muscle performance, it is necessary to determine the amount of energy contributed by the elastic tissues during movement. This, in turn, requires a quantification of the individual. time-varying forces and shortening velocities of the various elastic and contractile components of muscle. Elastic energy storage during jumping has been quantified in two recent studies, but the authors have arrived at opposing conclusions. Bobbert er al. (1986a) used an inverse-dynamics method to estimate the time histories of the musculotendon forces and shortening velocities in soleus and gastrocnemius during a maximum-height countermovement jump (CMJ) (i.e. a vertical jump involving significant downward motion of the center of mass of the body prior to upward propulsion). These researchers estimated that the elastic tissues of the ankle plantarflexors contributed up to 40% of the total energy delivered by these muscles during the ground contact phase of the jump. They concluded, therefore, that elastic energy storage is important to overall jumping performance. More recently, Pandy (1990) used a direct-dynamics method to compute the musculotendon forces, body-segmental motions, and muscle activation patterns for a series of maximum-height squat jumps (SJ) (i.e. vertical jumps beginning with the body in a static, semi-squatting position). He found that jumping performance is most sensitive to changes in body strength-to-weight ratio and muscle-fiber contraction speed, and is least sensitive to changes in tendon 1413 F. C. ANDERSONand M. G. 1414 compliance. Pandy (1990) concluded, therefore, in opposition to Bobbert et al. (1986a), that elastic energy storage in the tendons of the ankle plantarflexors is not a major determinant of jumping performance. ‘The above disagreement stems from the fact that both Bobbert et al. (1986a) and Pandy (1990) addressed only a subset of the questions which need to be answered in order to define the role of elastic tissues during jumping. In particular, neither study compared an SJ to a CMJ, and while Pandy (1990) did not estimate the amount of energy contributed by the elastic tissues, Bobbert et al. (1986a) did not estimate the energy contributed by many of the major lowerextremity muscles including vasti, hamstrings, and gluteus maximus. A major goal of this paper is to provide a better understanding of how the elastic tissues enhance muscle performance during vertical jumping. To this end, we have used an optimal control model previously developed by Pandy et al. (1990) to compute the optimal control solutions to a maximum-height CMJ and a maximum-height SJ. After confirming the ability of the model to replicate the major features of each jump, we analyzed each solution to quantitatively address the following questions: (1) What contributions do the elastic tissues make to the total energy delivered to the skeleton during a CMJ and an SJ? (2) Given that the extensor muscles undergo eccentric contractions during a CMJ, do any of these muscles develop larger forces and store more elastic energy during a CMJ than an SJ? (3) Is there a difference in the way elastic energy is stored and utilized during each jump? (4) How does tendon compliance affect the role of the elastic tissues during jumping? (5) Does the utilization of stored elastic strain energy lead to an increase in jump height? METHODS Experiments with human subjects Five strong, athletic, adult males (age 25+ 5 yr, height 183+3 cm, and body mass 78k5 kg) were chosen as subjects for these experiments. Each subject performed five CMJs and five SJs in alternating order. All jumps were performed with hands crossed over the chest to eliminate arm swing. For the CMJ, each subject began from a relaxed standing position and was instructed to ‘jump as high as possible’. For the SJ, each subject began from a deep squatting position and was instructed to ‘jump as high as possible, without countermoving’. Subjects were instructed not to countermove during the SJ in order to accentuate the difference between the SJ and the CMJ. For the SJ, each subject’s initial body position was chosen to be the lowest position of his preceding CMJ. The subject moved into this position by viewing himself on a large-screen television monitor and align- PANDY ing his body markers with a template superimposed on the television screen. Each subject-specific template was made by viewing a video of the subject’s CMJ frame by frame, locating the frame corresponding to the lowest position of the CMJ, and recording the location of the subject’s body markers in this position. For all jumps, force-plate, limb position, and electromyographic (EMG) data were recorded simultaneously. Ground reaction forces were measured using a six-component, strain-gauged force platform (Bertec Corp., Columbus). Fore-aft and vertical channels were sampled at 1000 Hz, as were the analog EMG data. Pairs of preamplified EMG surface electrodes (Iomed Inc., Salt Lake City) 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), rectus femoris (RF), hamstrings (HAMS), and gluteus maximus (GMAX). To record the joint angular displacements for each subject, retroreflective markers were positioned over five bony prominences: the head of the fifth metatarsophalangeal joint, the lateral malleolus, the lateral the greater trochanter, epicondyle, and the glenohumeral joint. Together, these landmarks defined the four body segments in the model: foot, shank, thigh, and HAT (head, arms, and trunk) (Fig. 1). Using a kinematic data acquisition system (Motion Analysis Inc., Santa Rosa), the absolute displacements of the markers were recorded at 60 Hz. Postprocessing of these data was carried out on a Silicon Graphics Personal Iris computer workstation. Experimental determination of jump height For each subject, we determined jump height directly from the measured vertical ground reaction force. Specifically, we used the vertical ground force to compute the vertical acceleration of the center ofmass of the body, and then numerically integrated this trajectory to obtain trajectories of the vertical velocity and vertical displacement of the whole-body center of mass during the ground contact phase of the jump. We then calculated jump height (J) using the vertical velocity ( f(tr)) and vertical displacement (Y(t,)) of the center of mass at lift-off (i.e. the instant the body leaves the ground): J= Y(rf)+ ?&)/2g, (1) where g is the gravitational acceleration constant. Because integrating the vertical acceleration and vertical velocity of the center of mass only produces a net change in position, for both the CMJ and the SJ we recorded the vertical ground reaction force with the subject beginning from a static, standing position. Specifically, for the SJ, we measured the vertical force as each subject descended from standing into a deep squat, and included the vertical acceleration gener- 1415 Utilization of elastic strainenergy GMAX 1.5 s, we estimate the error in jump height to be k 2 cm, or 3% of the jump height. For an SJ, with data collected for 4 s, we estimate the error in jump height to be k 6 cm, or 9% of the jump height. Data collection for an SJ took 4 s because, in addition to collecting data as the subject moved from the standing position into the prespecified squat, data were collected as the subject paused in the squatting position for l-2 s. Computation of optimal controls Fig. 1. Schematic representation of the musculoskeletal model used to simulate a maximum-height CMJ and a maximum-height SJ. The human skeleton was modeled as a four-segment, four-degree-of-freedom, planar linkage, joined to the ground at the toes and articulated at the ankle, knee, and hip by frictionless revolute joints. A total of eight musculotendinous units actuated the model: soleus (SOL), gastrocnemius (GAS), other plantarflexors (OPF, which represents the remaining uniarticular ankle extensors), tibialis anterior (TA), vasti (VAS), rectus femoris (RF), hamstrings (HAMS), and gluteus maximus (GMAX). To model the interaction of the foot with the ground, a highly damped, stiff, torsional spring was placed at the toes to provide a restoring torque whenever the heel dropped below ground level. Details of the model are given in Pandy et al. (1990). ated during the descent in our calculation of jump height. Thus, jump height for both the CMJ and the SJ was computed as the net change in vertical displacement of the center of mass from standing. To assess the accuracy of this method, we collected data as one subject countermoved from a relaxed standing position and then returned to his original upright position. In this way, the net vertical displacement of the subject’s center of mass, as determined from the force plate, could be compared to the motion actually executed by the subject (i.e. a zero net vertical displacement of the center of mass of the body). Error in the estimation of the subject’s center of mass position was found to accrue at a rate of 1.5 cm s-r for the duration of data collection. We attribute this error to the inability of the force plate to follow precisely the rapid changes in force applied to it (see Anderson, 1992, for details). For a CMJ, with data collected for Previously, Pandy et al. (1990) computed the optimal control solution to a maximum-height SJ by assuming that the optimal controls were bang-bang (i.e. either on or off). Because such an assumption may not be applicable to the downward phase of a CMJ, we have developed an algorithm for computing the non-bung-bung optimal controls appropriate to maximum-height jumping (Pandy et al., 1992). A key feature of this algorithm is the conversion of the optimal control problem into a parameter optimization problem. Converting the optimal control problem into a parameter optimization problem involves specifying the muscle excitation history for each muscle at discrete intervals of time control nodes. Values of muscle excitation at each of these control nodes form a set of unknown variables in the resulting parameter optimization problem. By linearly interpolating between the control nodes, provided that the spacing is sufficiently small, the continuous excitation histories for each of the muscles in the model can be reconstructed for the purpose of a forward integration of the equations of motion. Given a set of control nodes, each iteration of the algorithm consists of multiple forward integrations of the equations of motion to evaluate the performance criterion (i.e. jump height), the constraints (e.g. a zero vertical ground reaction force at lift-off), and the first derivatives of the performance criterion and the constraints with respect to each control. This information is then input to a parameter optimization routine (Powell, 1978) to find a new, improved set of controls (see Pandy et al., 1992, for details). To replicate the conditions under which our subjects executed the SJ, the optimal control problem for a maximum-height SJ included a constraint that required the vertical velocity of the center of mass of the model to remain positive throughout the jump. In this way, beginning from the lowest position of the CMJ, the model, like our subjects, was prevented from executing any preparatory countermovement during the SJ. RESULTS Comparison of model and experiment In general, there was good agreement between model and experiment for both jumps. For the CMJ 1416 F. C. ANDERSONand M. G. PANDK Displacement (deg) 160 140 120 loo 80 60 140 140 100 80 60 20 20 0 20 40 % Ground 60 80 100 Contact Time 0 20 % 40 60 80 100 Ground Contact Time Fig. 2.Joint angular displacements of the ankle, knee, and Fig. 3. Joint angular displacements of the ankle, knee, and hip during the ground contact phase of a maximum-height hip during the ground contact phase of a maximum-height CMJ for the model (thick grey lines) and the subjects (thin SJ for the model (thick grey lines) and the subjects (thin solid solid lines). The displacements predicted by the model lie lines). The displacements predicted by the model lie almost almost entirely within those measured for our subjects. Each entirely within those measured for our subjects. Each experimental trajectory corresponds to a subject’s highest experimental trajectory corresponds to a subject’s highest jump. For the model and the subjects, 0% of ground contact jump. 0% of ground contact time defines the instant that the time defines the instant that the vertical ground force vertical ground force changed by about 5% from body weight, while 100% defines the instant of body lift-off. decreased to about 95% of body weight, while 100% defines the instant that the body left the ground. For the model, prior to and at 0% of ground contact time, muscle forces the order hip, knee, and ankle) for both the CMJ and were computed to maintain the body in static equilibrium. and the SJ, the joint angular displacements predicted by the model were almost entirely within the range of joint angular displacements measured for the subjects (compare thick grey and thin solid lines in Figs 2 and 3). Peak vertical ground reaction forces predicted by the model for the CMJ and the SJ were also within the range measured for our subjects [cf. thick grey and thin solid lines in Figs 4(a) and S(a)]. In addition, both the model and the subjects generated similar trajectories of the vertical velocity and vertical displacement of the center of mass during the CMJ and the SJ [Figs 4 and 5(b) and (c)l. Experimental EMG activity agreed qualitatively well with the computed optimal controls for both jumps (Figs 6 and 7, compare thick solid lines with light wavy lines). In general, the model predicted a proximal-to-distal muscle activation sequence (i.e. in the SJ. Contrary to experiment, however, the model activated TA and RF early during the CMJ to accelerate the trunk into flexion (Fig. 6, TA and RF). These differences, we believe, are due to an attempt by TA and RF in our model to compensate for the absence of uniarticular knee and hip flexor muscles which act to accelerate the trunk downward during the preparatory countermovement (Anderson and Pandy, unpublished results). The model and our subjects left the ground at about the same time. The model took 1.04 s to leave the ground for the CMJ and 0.45 s for the SJ. By comparison, subject lift-off times ranged from 1.1 to 1.3 s for the CMJ and 0.46 to 0.64 s for the SJ. Finally, our subjects jumped on average 5% higher during the CMJ than they did during the SJ, although some subjects performed equally well during both jumps (see Table 1). The model, on the other hand, jumped 2% higher during the SJ than it did during the CMJ. 1417 Utilization of elastic strain energy Center of Mass Acceleration (mls’) 3o (O/O r Force body weight) 1 400 20 c Velocity (mls) Force (% body weight) Center of Mass Acceleration (mls2) a Velocity (mls) Displacement (m) Displacement(m) 0.2 r -0.6 I 20 0 % I 40 I 60 I 80 I 100 Ground Contact Time Fig. 4. Trajectories of the vertical ground reaction force and vertical acceleration (a). vertical velocity (b), and vertical displacement (c) of the center of mass of the body predicted by the model (thick grey lines) and those generated by the subjects (thin solid lines) during the ground contact phase of a maximum-height CMJ. The model and subjects generated vertical ground reaction forces which were roughly twice body weight, and trajectories of the vertical velocity and displacement of the center of mass which were similar. All displacement, velocity, and acceleration trajectories were derived directly from the vertical ground reaction force generated during the jump (see Methods). Energy delivered to the skeleton The elastic tissues delivered substantial amounts of energy to the skeleton during both the CMJ and the SJ. Tendon, the series-elastic elements (SEES), and the parallel-elastic elements (PEES) combined contributed 35% of the total energy delivered to the skeleton by all the musculotendon actuators [Fig. 8(a); add contributions from Tendon, the SEE and the PEE and compare with Total]. The contractile elements (CEs) accounted for the remaining 65% [Fig. 8(a), compare CE with Total]. Furthermore, the total energy delivered to the skeleton by all the musculotendon actuators, as well as the energy delivered by all the tendons, all the SEES, all the PEES, and all the CEs was almost the same for the CMJ and the SJ -0.6 1 0 I 20 I 40 I 60 I 80 I 100 % Ground Contact Time Fig. 5. Trajectories of the vertical ground reaction force and vertical acceleration (a), vertical velocity (b), and vertical displacement (c) of the center of mass of the body predicted by the model (thick grey lines) and those generated by the subjects (thin solid lines) during the ground contact phase of a maximum-height SJ. The model and the subjects generated vertical ground reaction forces which were roughly twice body weight, and trajectories of the vertical velocity and displacement of the center of mass which were similar. [Fig. 8(a); cf. grey and black shaded bars for Tendon, SEE, PEE, CE, and Total]. VAS and GMAX were the major energy producers of the lower extremity, followed by HAMS and the ankle plantarflexors. VAS, GMAX, and HAMS contributed 75% of the total energy delivered to the skeleton, while the ankle plantarllexors, SOL, OPF, and GAS, accounted for the remaining 25% [Fig. 8(b); height of bars for all actuators added]. RF did very little positive work on the skeleton during the propulsion phase of the jump [Fig. 8(b); RF]. The amount of energy delivered to the skeleton by an actuator was heavily influenced by the compliance of its tendon. For the more proximal muscles (i.e. GMAX, HAMS, and VAS), which have relatively short and stiff tendons, the total energy delivered to 1418 F. C. ANDERSON /T GMAX and M. G. PANDV GMAX RF RF VAS VAS OPF OPF /‘ SOL SOL GAS TA GAS ,A= TA 20 % I I I I I 20 40 60 80 100 % g!+-wb& 0 I 0 40 60 80 1W Ground Contact Time Fig. 6. Experimental EMG activity from one subject (thin wavy lines) and the optimal muscle excitation signals (controls) predicted by the model (thick solid lines) during the ground contact phase of a maximum-height CMJ. The muscle activation patterns predicted by the model agree qualitatively well with the measured EMG activity for each muscle. With the exception of TA and RF, muscles were activated in a proximal-to-distal sequence. No EMG activity was recorded from OPF because these are deep-lying muscles of the calf. The light horizontal lines indicate the zero level for both the EMG and the optimal controls. the skeleton was dominated by the contractile elements [Fig. 8(b); cf. shaded and empty bars for GMAX, HAMS, and VAS]. In contrast, for the ankle plantarflexors, which possess longer and more compliant tendons, the total energy delivered to the skeleton was dominated by the elastic tissues [Fig. 8(b); cf. shaded and empty bars for SOL, OPF, and GAS]. In fact, the elastic tissues accounted for almost 70% of the total energy delivered to the skeleton by the ankle plantarflexors. Optimal muscle forces Since stored elastic strain energy is directly proportional to muscle force, to assess the difference in the amount of elastic energy stored during the CMJ and the SJ, we examined the peak forces developed by all the muscles during each jump. Of the eight lowerextremity muscles included in the model, only VAS and HAMS developed much more force during the CMJ than they did during the SJ. The force in VAS was about 1000 N greater, and the force in HAMS was about 500 N greater (cf. VAS and HAMS in Figs 9 and 10). VAS and HAMS were able to generate Ground Contact Time Fig. 7. Experimental EMG activity from one subject (thin wavy lines) and the optimal muscle excitation signals (controls) predicted by the model (thick solid lines) during the ground contact phase of a maximum-height SJ. The muscle activation patterns predicted by the model agree qualitatively well with the measured EMG activity for each muscle. In general, muscles were recruited proximally to distally. No EMG activity was recorded from OPF. Table 1. SJ height, CMJ height, and the ratio of SJ height to CMJ height for the subjects and the model. Subject SJ (cm) CMJ (cm) SJjCMJ 1 2 3 4 5 6 7 8 9 10 11 12 13 Model 43 45 45 45 46 47 51 54 55 53 54 61 63 65 45 46 46 49 50 50 52 54 57 58 60 61 68 64 0.96 0.98 0.98 0.92 0.92 0.94 0.98 1.00 0.97 0.91 0.90 1.00 0.93 1.02 Note: To improve our estimate of the difference in performance between the CMJ and the SJ, we measured jump height for eight additional subjects, each of similar size and athletic ability to the five subjects originally chosen for this study (see Methods). Subjects generally jumped higher during the CMJ. However, the difference in performance between the two jumps was typically small. Two subjects performed equally well during both jumps, and more than half of our subjects jumped less than 5% higher during the CMJ than the SJ. In contrast, the modef jumped 1 cm higher during the SJ. Utilization of elastic strain energy Joules 1419 Force (N) 800 CE 600 a i Joules 300 VAS t ill OPF 3000 - GMAX rk 2000 TA 0 20 40 60 80 100 % Ground ContactTime b Fig. 8. (a) Total positive work done on the skeleton by all the musculotendon actuators except TA (TOTAL), all the parallel-elastic elements (PEE), all the tendons (Tendon), all the series-elastic elements (SEE), and all the contractile elements (CE) during the SJ (grey bars) and the CMJ (black bars). Note that the work performed on the skeleton by an actuator element is not necessarily the same as the total work performed by that element (see Appendix A). The contractile elements contributed 65% of the total energy delivered to the skeleton. while the elastic tissues accounted for the remaining 35%. Notice also that all the musculotendon actuators. as well as all the elements of all the actuators, contributed nearly the same amount of energy to the skeleton during both jumps. (b) Total positive work done on the skeleton by the contractile elements (shaded bars) and the elastic tissues (empty bars) of each muscle except TA during the SJ and the CMJ (first and second histogram, respectively). The elastic tissues of the ankle plantarflexors (SOL, GAS, and OPF) account for about 70% of the energy delivered to the skeleton by these actuators, while the contractile elements of the more proximal actuators (VAS, HAMS, and GMAX) dominated the energy delivered by these muscles during both the CMJ and the SJ. Note that each actuator, the contractile element of each actuator, and the elastic tissues of each actuator contribute nearly the same amount of energy to the skeleton during both jumps. Fig. 9. Optimal muscle forces predicted by the model plotted against ground contact time for a maximum-height CMJ. The vertical dotted line at about 70% of ground contact time represents the instant that the center of mass of the model began to move upward. Notice that the peak forces in the more proximal muscles, VAS, HAMS, and GMAX, occurred before the center of mass of the model began moving upward, whereas the peak forces in the ankle plantarflexors, SOL, OPF, and GAS, occurred during the propulsion phase of the jump. Only VAS and HAMS developed much more force during the CMJ than they did during the SJ (compare with forces in VAS and HAMS for the SJ in Fig. 10). As a result, these muscles stored more elastic strain energy during the CMJ. However, since VAS and HAMS have relatively stiff tendons, the increase in stored elastic energy during the CMJ was not large. the increase in stored elastic energy during the CMJ was not large. For example, the 1000 N increase in VAS force resulted in an increase of only 7 Joules of stored elastic strain energy in the tendon of VAS. This, together with the fact that the ankle plantarflexors developed about the same amount of force during each jump, indicates that there was almost as much elastic strain energy stored during the SJ as was stored during the CMJ. more force during the CMJ because they were fully Mechanisms of energy transfer activated while undergoing eccentric contraction during the downward phase of the jump. In contrast, all of the ankle plantarflexors were fully activated only after they had begun concentrically contracting during the propulsion phase of the CMJ. Because of this coordination, the plantarflexors developed more or less the same amount of force during the CMJ and the SJ (Figs 9 and 10, cf. SOL, OPF, and GAS). Because VAS and HAMS developed more force during the CMJ than the SJ, these muscles stored more elastic strain energy during the CMJ. However, since VAS and HAMS have relatively stiff tendons, To examine in detail how the elastic tissues store and transfer energy during a CMJ and an SJ, for each jump we calculated the amount of energy lost as heat [Fig. 1l(a)], the amount of gravitational potential energy that was converted into elastic strain energy [Fig. 11(b)], and the amount of elastic strain energy created by the contractile elements [Fig. 11(c)]. In interpreting the results which follow, it is essential to distinguish between the total energy output of an element of an actuator and the energy delivered by that element to the skeleton. For example, the contractile elements can deliver energy to the elastic tissues at 1420 F. C. ANDERSONand M. G. PANDY Force (N) Joules a Joules 80 b 60 Joules 60 01 0 TA 1----,-e--___ 20 40 60 40 ___ 80 GMA 100 % Ground Contact Time Fig. 10. Optimal muscle forces predicted by the model plotted against ground contact time for a maximum-height SJ. The peak muscle forces developed by the ankle plantarflexors, SOL, OPF, and GAS, were about the same during the SJ and the CMJ (compare with forces in SOL, OPF, and GAS for the CMJ in Fig. 9). the same time as they deliver energy to the skeleton (see Appendix A). Our calculations indicate that much more energy was lost as heat during the CMJ than the SJ [Fig. 11(a); cf. grey and black shaded bars]. In fact, about 50% of the gravitational potential energy initially available in the CMJ was lost as heat in the contractile elements of GMAX, RF, and VAS as these muscles developed large forces to accelerate the trunk upward during the downward phase of the jump (Fig. 9, VAS, HAMS. GMAX). (We note here that there were almost 300 J of gravitational potential energy initially available in the CMJ that were not available in the SJ since the center of mass of the model began from a higher position in the CMJ than it did in the SJ.) Our calculations also indicate that much more gravitational potential energy was converted into elastic strain energy during the CMJ than the SJ [Fig. 11(b); cf. grey and black shaded bars]. Again, the muscles which showed the largest differences between the two jumps were the more proximal extensors [Fig. 11(b), GMAX, HAMS, and VAS]. We believe that this conversion of gravitational potential energy into elastic strain energy during the CMJ leads to a more efficient jump (see Discussion). Finally, the contractile elements not only delivered the same amount of energy to the skeleton during both jumps, but they also created large amounts of elastic strain energy during the SJ [Fig. 11(c); com- 20 0 Fig. 11. (a) Energy lost as heat in the contractile elements of each muscle except TA during the SJ (grey bars) and the CMJ (black bars). Much more energy was lost during the CMJ than during the SJ in the contractile elements of VAS, RF, and GMAX. These muscles accelerated the trunk upward during the countermovement phase of the CMJ, during which time the contractile elements were stretched by the downward motion of the body segments. (b) Energy transferred from the skeIeton to the elastic tissues of each muscle except TA during the SJ (grey bars) and the CMJ (black bars). During the CMJ, significant gravitational potential energy was stored as strain energy in the elastic tissues of VAS, HAMS, and GMAX. (c) Energy delivered by the contractile elements to the elastic tissues of each muscle except TA during the SJ (grey bars) and the CMJ (black bars). Much more energy was delivered by the contractile elements to the elastic tissues during the SJ than during the CMJ, especially in VAS and GMAX. The total work performed by the contractile elements during the SJ can be found by adding all the grey bars here and in Fig. 8(b). Similarly, for the CMJ, add all black bars here and in Fig. 8(b). pare grey and black shaded bars for all the muscles, especially VAS and GMAX]. By performing additional work during the SJ, the contractile elements compensated for the amount of gravitational potential energy that was converted into elastic strain energy during the CMJ [Fig. 11(b)]. Effect of tendon compliance To determine the effect of tendon compliance on elastic energy storage during jumping, we altered the compliance of VAS, RF, HAMS, and GMAX in our model until the strain in each tendon became 10% (i.e. the maximum strain defining tendon rupture). We chose these actuators because they have relatively stiff tendons, and because they dominated the total energy Utilization 1421 of elastic strain energy height was only 3% higher for both the CMJ and the SJ when tendon compliance was increased to its limit. a CE Joules 400 VAS 200 100 0 Fig. 12. Energy delivered to the skeleton when tendon comohance in VAS. RF. HAMS. and GMAX was increased to the point where the strain in tendon became 10%. (a) Total positive work done on the skeleton by all the musculotendon actuators except TA (TOTAL), all the parallel-elastic elements (PEE), all the tendons (Tendon), all the series-elastic elements (SEE), and all the contractile elements (CE) during the SJ (grey bars) and the CMJ (black bars). Increasing tendon compliance did not significantly alter the total energy delivered to the skeleton during either jump [cf. TOTAL with Fig. 8(a)]. The increase in energy delivered by the elastic tissues was counteracted by a decrease in the contribution from the contractile elements. Note. however, that differences in the amount of energy delivered by the elastic tissues and the contractile elements during the SJ and the CMJ are now more pronounced [cf. with Fig. 8(a)]. (b) Total positive work done on the skeleton by the contractile elements (shaded bars) and the elastic tissues (empty bars) of each muscle except TA during the SJ and the CMJ (first and second histogram, respectively). Increasing tendon compliance led to a substantial increase in <he amount of energy delivered by VAS, esoeciallv during the CMJ fcf. with Fig. 8(b)l. Note that the elastic;issues of VAS delivered much more energy to the skeleton during the CMJ than they did during the SJ. to the skeleton during both jumps [see Fig. 8(b)]. Increasing tendon compliance in the model led to an increase in the amount of energy delivered by the elastic tissues to the skeleton during both jumps. The energy delivered by the elastic tissues to the skeleton increased by 80% for the CMJ and by 50% for the SJ [Figs 8(a) and 12(a); cf. Tendon, PEE, and SEE with Total]. However, increasing tendon compliance also led to a decrease in the amount of energy delivered by the contractile elements to the skeleton [Figs 8(a) and 12(a); cf. CE with Total]. This explains why our jump delivered DISCUSSION A detailed analysis of the optimal control solutions to a maximum-height CMJ and a maximum-height SJ has provided us with considerable insight into how muscles and tendons store and deliver energy to the skeleton during the ground contact phase of jumping. In particular, knowledge of the individual forces and shortening velocities of the various elastic and contractile components of the major muscles in the lower extremity has enabled us to quantitatively address the following questions: (1) What contributions do the elastic tissues make to the total energy delivered to the skeleton during a CMJ and an SJ? The most surprising prediction made by our model was that the elastic tissues deliver nearly the same amount of energy to the skeleton during the CMJ and the SJ. During both jumps, the elastic tissues contributed about 35% of the total energy delivered to the skeleton [Fig. 8(a)]. This result can be understood by examining the peak muscle forces developed during each jump (question No. 2 below), and by quantifying the way in which mechanical energy was transferred between the elastic tissues and the contractile elements during the CMJ and the SJ (question No. 3 overleaf). (2) Given that the extensor muscles undergo eccentric contractions during a CMJ, do any of these muscles develop larger forces and store more elastic energy during a CMJ than an SJ? Of the eight lower-extremity muscles included in the model, only VAS and HAMS developed much larger forces during the CMJ (Figs 9 and 10). The plantarflexors (SOL, OPF, and GAS) did not develop larger forces during the CMJ because they were maximally activated only after they had begun concentrically contracting during the propulsion phase of the jump. This predicted coordination of the plantarflexors means that only the more proximal extensor muscles stand to benefit from countermovement. Unfortunately, the more proximal muscles (GMAX, VAS, and HAMS) have tendons that are relatively short and stiff. As a result, an increase in muscle force in the more proximal extensors did not result in a large increase in the amount of elastic energy stored during the CMJ [Fig. 8(b), GMAX, VAS, and HAMS]. Because the peak muscle forces do not yield information about how elastic strain energy was utilized during each jump, the above results do not completely explain why the energy delivered by the elastic tissues to the skeleton was nearly the same for both jumps. What they do indicate is that in our model there was F. C. ANDERSONand M. G. PANDY 1422 little difference in the amount of energy stored by the elastic tissues during the SJ and the CMJ. (3) Is there a differPnce in the way elastic energy is stored and utilized during each jump? In general, elastic strain energy can originate from two sources: (i) from the kinetic or gravitational potential energy of the skeleton, and (ii) from the contractile elements which convert chemical energy into mechanical energy. In our model, a large portion of the energy stored in the elastic tissues during the CMJ came from the gravitational potential energy of the skeleton as the more proximal extensor muscles were stretched during the downward phase of the jump [Fig. 11(b)]. During the SJ, on the other hand, strain energy stored in the elastic tissues came primarily from the contractile elements as they did work to stretch the tendons and the SEES of the muscles [Fig. 11(c)]. The fact that the contractile elements delivered approximately the same amount of energy to the skeleton during both jumps [Fig. 8(a)], together with the fact that the contractile elements delivered more energy to the elastic tissues during the SJ [Fig. 11(c)], means that the contractile elements actually performed more total work during the SJ than the CMJ. Once stored, elastic strain energy can be delivered either to the skeleton or to the contractile elements. If energy is delivered to the contractile elements, it is lost as heat. In our model, relatively little energy was lost as heat during the SJ, which means that most of the energy stored in the elastic tissues was delivered to the skeleton. In contrast, during the CMJ, a larger portion of the stored elastic strain energy was delivered to the contractile elements and dissipated as heat [Fig. 11(a), cf. grey and black bars]. The above results provide a quantitative and complete explanation for why the energy delivered by the elastic tissues to the skeleton was nearly the same for the CMJ and the SJ. The explanation is two-fold. First, by performing more total work during the SJ, the contractile elements were able to create nearly as much elastic strain energy during the SJ as was stored during the CMJ. This additional work performed by the contractile elements during the SJ compensated for the large amount of gravitational potential energy that was converted into elastic strain energy during the CMJ. Second, even though a little more elastic strain energy was stored during the CMJ, this additional energy was lost as heat. (4) How does tendon compliance affect the role of the elastic tissues during jumping? In general, the proportion of energy delivered by an actuator’s elastic tissues to the skeleton is largely determined by tendon compliance. For example, in our model, the elastic tissues of the ankle plantarflexors, which have relatively long and compliant tendons, contributed 70% of the total energy delivered by these muscles [Fig. 8(b)]. In contrast, for the more proximal muscles such as GMAX, HAMS, and VAS, which have relatively short and stiff tendons, the contractile elements dominated the total energy delivered to the skeleton [Fig. 8(b)]. Increasing tendon compliance in the model led to a significant increase in the proportion of energy delivered by the elastic tissues to the skeleton, but it did not result in a significant increase in the total amount of work done on the skeleton (Fig. 12). This explains why in our model jump height for both jumps increased by only 3% when tendon compliance was increased to its limit. (5) Does the utilization of stored elastic strain energy lead to an increase in jump height? Given the fidelity of our optimal control model for jumping (see below), we cannot resolve whether the participation of the elastic tissues is responsible for the experimentally measured difference in jump height between a CMJ and an SJ. What is clear, however, is that the participation of the elastic tissues leads to differences between a CMJ and an SJ which are more striking than the relatively small difference in jump height. Our model predicts that a significant amount of energy is lost as heat during a CMJ, that a significant amount of gravitational potential energy is converted into elastic strain energy during a CMJ, and that the contractile elements perform more total work during an SJ than they do during a CMJ. These results lead us to conclude that when propulsion is preceded by a preparatory countermovement, storage and utilization of elastic strain energy leads to a more efficient jump rather than a significantly higher jump. Limitations of the model The credibility of our results rests heavily upon the validity of our optimal control model for jumping. The fact that our model was able to reproduce the pattern of body-segmental motions, ground reaction forces, and muscle activations for both the CMJ and the SJ is strong evidence that it is accurate enough to replicate the major features of each jump. There are, however, limitations of our optimal control model which require further scrutiny. In particular, our model does not include any of the uniarticular muscles spanning the knee and the hip, such as the short head of biceps femoris and iliopsoas. As a result, the optimal control solution for the CMJ was probably not as well coordinated as it would have been had these muscles been included in the model. However, since the energy delivered to the skeleton during the CMJ and the SJ is dominated by the hip and knee extensors, GMAX, HAMS, and VAS [Fig. 8(b)], it is unlikely that our conclusions would be altered by adding uniarticular flexor muscles at the hip and knee. There are also limitations associated with our model for muscle. First, our model for muscle neglects the effects of muscle mass. As a result, tendon and the Utilization of elastic strain energy SEES of muscle behave as springs attached to a massless force generator, the contractile element. However, since the inertial force associated with muscle mass is much smaller than muscle’s maximum isometric force, we do not expect the addition of muscle mass to alter the results of our analyses significantly. Second, although the shape of the force-velocity curve assumed by our model closely follows the force-velocity curve derived from isolated muscle experiments (Katz, 1939), our model’s force-velocity curve does not precisely reproduce the sudden increase in contractile element force observed experimentally as muscle makes its transition from shortening to lengthening. Improving this aspect of our muscle model could lead to more noticeable differences between the amount of elastic strain energy stored during the CMJ and the SJ. However, because our model predicts that significant gravitational potential energy is converted into elastic strain energy during the CMJ and that the contractile elements create large amounts of elastic strain energy during the SJ, we expect the conclusion that the elastic tissues affect jumping efficiency more than jumping performance to remain unchanged. Limitations of the experiments Much interest in the storage and utilization of elastic strain energy has been fueled by experiments which have shown that humans typically jump higher during a CMJ than they do during an SJ. Notably, Asmussen and Bonde-Petersen (1974) and Komi and Bosco (1978) found CMJs to be approximately 5 and 10% higher than SJs, respectively. Although our experimental estimates of the difference in jump height between a CMJ and an SJ agree with the difference published by Asmussen and Bonde-Petersen (1974), it is also evident from our work that several problems exist which make an accurate comparison of jump heights difficult. The first of these is simply that the difference in jump height is small, perhaps smaller than the errors associated with the experimental methods used. We found the mean difference in jump height between the two jumps to be 5% (Table l), but we estimated the errors in the jump height measurements for the CMJ and the SJ to be + 3 and f 9%, respectively. There is also reason to believe that timeof-flight methods (Asmussen and Bonde-Petersen, 1974; Komi and Bosco, 1978) and kinematic methods (Bobbert et al., 1986b) for measuring subject jump height possess similar magnitudes of error. For example, Fukashiro et al. (1983) estimated the accuracy of their jump height calculations to be +4% by a comparison of film data with forceplate and goniometer data. It is difficult, therefore, to resolve with accuracy the true difference in jump height between a CMJ and an SJ. A second confounding factor involves the position from which upward propulsion begins during the SJ, a consideration which may lead to systematic error in 1423 the comparison of the two jumps. As discussed in detail in Appendix B, we believe that the body position from which upward propulsion begins is a major determinant of jumping performance. Therefore, in studies in which the initial position of the SJ was not controlled, differences in jump height for the CMJ and the SJ may have arisen not from the utilization of stored elastic strain energy, but rather from differences in the body position from which upward propulsion began. Unfortunately, specifying the initial position of the SJ is not without problems either. Many of our subjects found it difficult to assume the prespecified limb angles for the SJ because of the strength and balance required by the initial deep squatting position. Therefore, our subjects may not have jumped as high during their SJs simply because of the difficulties associated with coordination and not because of a lesser utilization of stored elastic strain energy. Due to the problems surrounding the experimental determination ofjump height, we do not know how to assess the prediction made by our model that the SJ is 1 cm higher than the CMJ, other than to conclude that the two jumps are very similar in terms of jump height and, therefore, in terms of the total work done on the skeleton by all the lower-extremity muscles. Concluding remarks It is a complex undertaking to investigate in detail how the utilization of stored elastic strain energy enhances muscle performance during movement. A quantification of the energy contributed by the elastic tissues requires an accurate knowledge of the individual, time-varying forces and shortening velocities of all the various elements comprising muscle. The central complicating factor is that, in addition to delivering energy directly to the skeleton, the elements comprising muscle also have the ability to deliver energy to one another (see Appendix A for details). This fact has two lasting implications. First, stored elastic strain energy can be lost as heat if the elastic tissues perform work to stretch the contractile elements instead of moving the skeleton; this mechanism of energy loss was evident during the CMJ. Second, the contractile elements can themselves create significant amounts of elastic strain energy, a point which is vital to understanding how the elastic tissues were able to deliver nearly as much energy to the skeleton during the SJ as they did during the CMJ. Although aspects of our optimal control model for jumping could be improved, the model has enabled us to make a number of specific predictions regarding the role of elastic tissues during jumping, many of which agree with the findings of other workers. In agreement with the analyses conducted by Bobbert et al. (1986a), our model predicts that the elastic tissues of the ankle plantarflexors deliver large amounts of energy to the skeleton during a CMJ. However, the model also predicts the same to be true 1424 F. C. ANDERSON and M. G. PANDY for an SJ. In both the CMJ and the SJ, the elastic tissues of the plantarflexors accounted for 70% of the energy delivered to the skeleton by these actuators, and when all actuators were included, the elastic tissues accounted for 35% of the total energy delivered to the skeleton. These results suggest that countermovement does not significantly alter the amount of elastic strain energy which is stored and delivered to the skeleton during jumping. In agreement with Pandy (1990), we found that tendon compliance does not affect jump height significantly. For both the CMJ and the SJ, increasing tendon compliance to extremes produced only a 3% increase in jump height. The increase in tendon compliance did augment the amount of energy delivered to the skeleton by the elastic tissues, but it simultaneously decreased the amount of energy delivered by the contractile elements. This result supports the position forwarded by Cavagna (1977) that the elastic tissues do not significantly increase the total positive work done by muscles during jumping. Finally, our model provides evidence that substantial amounts of gravitational potential energy are stored and utilized during a CMJ and that the contractile elements perform more total work during the SJ than they do during the CMJ. Therefore, in agreement with results reported by Alexander and BennetClark (1977), Asmussen and Bonde-Petersen (1974b), KyrGlHinen et al. (1990), Morgan et al. (1978), and Taylor and Heglund (1982), we conclude that the elastic tissues serve to effect a more efJicient conversion of musculotendon energy into translational kinetic and potential energy of the skeleton. In the context of jumping, our findings further suggest that elastic energy storage and utilization affects efficiency much more than jump height. Acknowledgments-We thank Lawrence Abraham, Duane Knudson, Jim Ziegler, David Carpenter, and Kristin Daigle for reviewing an earlier version of this manuscript. We also acknowledge Jim Ziegler for his help with the jumping experiments. This work was supported by the Whitaker Foundation and NASA/Ames Research Center, Grant NCA2-532. REFERENCES Alexander, R. McN. and Bennet-Clark, H. C. (1977) Storage of elastic strain energy in muscle and other tissues. Nature 265, 114-117. Anderson, F. C. (1992) Storage and utilization of elastic strain energy during human jumping: an analysis based on the predictions of an experimentally verified optimal control model. Masters thesis, Department of Kinesiology and Health Education, University of Texas at Austin, Austin, TX. Asmussen, E. and Bonde-Petersen, F. (1974a) Storage of elastic energy in skeletal muscles in inan. Acta Physiol. stand. 91, 385-392. Asmussen, E. and Bonde-Petersen, F. (1974b) Apparent efficiency and storage of elastic energy in hum& muscles during exercise. Acta Physiol. Stand. 92, 537-545. Bobbert, M. F., Huijing, P. A. and van Ingen Schenau, G. J. (1986a) 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., Mackay, i., Schinkelshoek, D., Huijing, P. A. and van Ingen Schenau, G. J. (1986b) Biomechanical analysis of drop and countermovement jumps. Eur. J. appl. Physiol. 54, 566-573. Cavagna, G. A. (1977) Storage and utilization of elastic energy in skeletal muscle. Exercise and Sport Sciences Reviews 5, 89-129. Cavagna, GA., Dusman, B. and Margaria, R. (1968) Positive work done by a previously stretched muscle. J. appl. Physiol. 24, 21-32. Fukashiro, S., Ohmichi, H., Kanehisa, H. and Miyashita, M. (1983) Utilization of stored elastic energy in leg extensors. In Biomechanics (Edited by Matsui, H. and Kobayashi, K.), Vol. VIII-A, pp. 258-263. Human Kinetics Publishers, Champaign, IL. Ingen Schenau, G.J. van (1984) An alternative view of the concept of utilization of elastic energy in human movement. Hum. Mumt Sci. 3, 301-336. Katz, B. (1939) The relation between force and speed in muscular contraction. J. Physiol. %, 45-64. Komi, P. V. and Bosco, C. (1978) Utilization of stored elastic energy in leg extensor muscles by men and women. Med. Sci. Sports Exercise 10, 261-265. Kyriillinen, H., Komi, P. V., Oksanen, P., Hgkkinen, K., Cheng, S. and Kim, D. H. (1990) Mechanical efficiency of locomotion in females during different kinds of muscle action. Eur. J. appl. Physiol. 61, 446-452. Levine, W. S., Christodoulou, M. and Zajac, F. E. (1983) On propelling a rod to a maximum vertical or horizontal distance. Automatica 19, 321-324. Morgan, D. L., Proske, U. and Warren, D. (1978) Measurements of muscle stiffness and the mechanism of elastic storage of energy in hopping kangaroos. J. Physiol. 282, 253-261. Pandy, M. G. (1990) An analytical framework for quantifying muscular action during human movement. In Multiple Muscle Systems: Biomechanics and Movement Organization (Edited by Winters, J. M. and Woo, S. L.-Y.), pp. 653-662. Springer, New York. Pandv, M. G., Zaiac, F. E., Sim, E. and Levine, W. S. (1990) An-optimal control model for maximum-height huma; iumnine. J. Biomechanics 23, 1185-1198. Pandy: I% G., Anderson, F. .C. and Hull, D. G. (1992) A parameter optimization approach for the optimal control of large-scale musculoskeletal systems. J. biomech. Engng 114,450-460. Powell, M. J. D. (1978) A fast algorithm for nonlinearly constrained optimization calculations. In Numerical Analysis: Lecture Notes in Mathematics, (Edited by Matson, G. A.), Vol. 630, pp. 144-157. Springer, New York. Taylor, C. R. and Heglund, N. C. (1982) Energetics and mechanics of terrestrial locomotion. Annu. Rev. Physiol. 44,97-107. Zajac, F. E. (1989) Muscle and tendon: Properties, models, scaling, and application to biomechanics and motor control. In CRC Critical Reoiew of Biomedical Engineering (Edited by Bourne, J. R.), Vol. 17, pp. 359-411. CRC Press, Boca Raton. APPENDIX A Musculotendon energy delivered to the skeleton The equations used to calculate the power delivered to the skeleton by a musculotendon actuator and by its individual elements are summarized below. The energy delivered to the skeleton by each element of an actuator was computed by integrating each element’s power-time curve from the time Utilization 1425 of elastic strain energy the joint angular displacements, joint angular velocities, actuator forces, and muscle activations at each time step (see Anderson, 1992 for details). Using the computed joint angular displacements and velocities, each actuator’s length and shortening velocity were determined at each instant during the jump. Using the computed actuator force (FMT), actuator shortening velocity (VMT). and actuator length (LMT). the force and shortening velocity of the actuator’s T. M, PEE, SEE. and CE were calculated from the following equattons: I IC Tendon Actuator I I =FT=FMT, FM. (A]) Fig. 13. Schematic representation of the musculotendon model used in this study. Each actuator was modeled as a three-element. lumped-parameter muscle in series with tendon. The mechanical behavior of muscle was described by a Hill-type contractile element (CE) which modeled its force-length-velocity property, a series-elastic element (SEE) which modeled its short-range stiffness, and a parallel-elastic element (PEE) which modeled its passive properties. Tendon was represented by a linear stress-strain curve. the actuator began to shorten to the instant of lift-off. Therefore, the energy delivered to the skeleton represents the amount of positive work done on the skeleton during the ground contact phase of each jump. These calculations were carried out in order to assess whether or not a preparatory countermovement leads to an increase in the total positive work done on the skeleton by all the musculotendon actuators. In this respect, computing the energy delivered to the skeleton from the time the center of mass begins to move upward is inappropriate, since not all the extensor muscles begin shortening at the same time. Each musculotendon actuator (MT) was modeled as a three-element muscle (M) in series with tendon (T). The three elements of muscle were the series-elastic element (SEE), the parallel-elastic element (PEE), and the contractile element (CE) (see Fig. 13). The details of our musculotendon model are given in Appendix 2 of Pandy et al. (1990) (see also Zajac. 1989). Some definitions are: i = MT, T, M. PEE. SEE, or CE. F’= force exerted by the ith element of a musculotendon actuator. I” =contraction velocity of the ith element (a positive value indicates shortening). Plotal,L= total power output of the ith element. P,,,,,,=power delivered by the ith element to the skeleton (A positive value indicates that power is delivered from the element to the skeleton, while a negative value indicates that power is transferred from the skeleton to the ith element.) In calculating the energy delivered to the skeleton by a particular element of an actuator, it is important to account for the fact that in addition to delivering energy to the skeleton. an element can deliver energy to the other elements of the actuator. For this reason, the energy delivered by an element to the skeleton is not necessarily as great as the total energy released by that element. An isometric contraction is a good example of this scenario. Although the energy delivered to the skeleton during an isometric contraction is zero, energy is transferred between the contractile element and the elastic tissues as the force developed by the contractile element changes. Thus, in order to calculate the energy delivered fo rhe .ske/eton by a particular element, the time histories of the forces and shortening velocities of all the elements comprising the actuator must be known. The forces and shortening velocities of the individual elements comprising an actuator were calculated during a forward integration of the dynamical equations describing the jumping model. A forward integration yields values of p”t=FM_FFPEE IA3) Fct = Fstt iA4) 1 dFMT t,.T= __-~ kT dt ’ lA5) .(VM’= C’PEE = ].‘I). (A6) ,,a 1A7) VcE= { From the force-length-velocity curve for muscle] [See Appendix 2 in Pandy rr al. (199O)l. f/SEE= FM_ f/et (A8) (A9) where w is the width of the muscle, rc is the fiber length corresponding to the muscle’s maximum isometric strength, If is tendon slack length, kT is tendon stiffness. and Cl and C2 are constants given in Pandy er al. (1990). Total power output The total power output of the ith element is given by the product of the force and velocity of that element: P,O,s,,j= F’V’. (A 10) Power delivered to the skeleton Musculotendon actuator. Under all conditions, the power delivered to the skeleton by an actuator is the same as the total power output of the actuator: P~M.MT= Pmta,.~~=FMT VMT. (A]]) This result does nor hold for an element of an actuator because it is possible for that element to deliver power to another element of the actuator. The whole actuator, on the other hand, can only deliver power to the skeleton. 1426 F. C. ANDERSONand M. G. PANDY Muscle and tendon APPENDIX B Muscle and tendon deliver power to the skeleton under the following conditions: if {V”BO and VT&O} or {V”<O and VT<O}, then P .te,,T=FTVT and P .w,,=F~V~; 6412) if V”<O and VT>0 and VMT>O, then P skcl,T=FMTVMT and P ,kcl.M=O; 6413) if V”<O and VT>0 and VMT<O, then P skcl.T=O and Pskc,,M= FMTVMT; 6414) if V”>O and VT<0 and VMT>O, then Pske,,T=O and PSkc,,M=FMTVMT; Dependence of jumping performance and propulsion time on vertical acceleration When the compliance of VAS, RF, HAMS, and GMAX in our model was increased, we found that more energy was stored in and delivered by the elastic tissues to the skeleton during a CMJ than during an SJ (Fig. 12(a), Tendon and SEE). This suggests that a preparatory countermovement may enhance jumping performance somewhat. The question is how much? Even if the additional elastic energy storage elicited by a preparatory countermovement can explain the measured 5% increase in jump height during a CMJ (Table l), in terms of overall jumping performance such changes are relatively minor. To understand how large changes in jumping performance can be elicited by a preparatory countermovement, we present the following heuristic analysis. Central to the ideas proposed here is an analytical result based upon a solution to the optimal control problem for propelling a baton to 6415) if V”>O and VT<0 and VMT<O, then P ,kcl,T= F”‘VMT and P skcl.M --0. Parallel-elastic (‘416) element With Pskcl,M computed froFpap, Pskcl.PEE=~ Contractile Vertical Accekhtion (mls 1) 100 [ Pskcl,PEEis given by Pakcl.M~ element and series-elastic 6417) element If {{V CE~O and VSEE<O} or {VCE>O and VSEEaO}} and VTfO} or {V”>O and VT>O}], then and {{V”<O FCEVCE P.kel,CE =- Pskcl.SEE =- F”VM F”VM and Pske, M ’ b ,.=.$J 6418) Pskel.M~ ‘. : . ‘. If VCE>O and VSE”<O and Prkc,.M<O, then P sk.l,CE=” P rkel.SEE -p and t.419) ske,,M -Pskcl,PEE. If VCE>O and VsE’<O and PSk,,,M>O, then Pskcl.CE=P~krl,M-Pskcl.PEE -10 and P rkel.SEE -0 - (A20) If VCE<O and VSEE>O and Pakcl,M<O, then P skcl.CE -- p ake1.M -Pskcl.PEE and Pskcl.SEE -0 - (A21) If VCEcO and VSEE>O and Pskcl,M>O, then P skcl.CE =O Prkel.S.EE --p and 6422) sksl.M -Pskc,.PEE. In all cases, the power delivered to the skeleton by an actuator is equal to the sum of the powers delivered to the skeleton by all the actuator’s elements. P skc,.MT = Pskcl.T+ Pskcl,PEE + pskcl,SEE 6423) + Pskcl.CE. This fact is noted to emphasize that equations (AlO)-(A22) produce consistent results. Finally, the energy delivered to the skeleton by the ith element over the time interval [to, t,] is found from E rkcl.i = tf P skcl. i’ to 6424) 1 I ’ ’ -0.4 3 -0.2 3 ’ 0 ’ 1 I 0.2 Vertical Displacement (m) Fig. 14. Vertical acceleration of the center of mass of the model as a function of its vertical displacement with the body in different positions ranging from a deep squat to standing. Zero vertical displacement of the center of mass signifies standing. (a) Vertical acceleration of the center of mass vs the vertical displacement of the center of mass (i) when peak, isometric, extensor torques are applied at all the joints simultaneously (Peak, thin solid line), (ii) when maximum, isometric, extensor torques, accounting for muscle moment arms and force-length properties, are applied at all the joints simultaneously (Max, thin dashed line), and (iii) for a maximum-height CMJ and SJ (thick solid and dashed lines, respectively). For cases (i) and (ii), the vertical displacement of the center of mass of the model was obtained from limb angular displacements generated during a maximum-height SJ. (b) Vertical acceleration vs vertical displacement of the center of mass of the model for a maximum-height CMJ (solid line) and SJ (dashed line) when the model begins with its center of mass only 10 cm below standing. Note that even though the model generates much larger vertical accelerations during the SJ, jump height is 17 cm lower for the SJ than for the CMJ. Utilization of elastic strain energy a maximum vertical distance (Levine et al., 1983). Specifically, depending upon the initial angle of the baton, two types of control have been shown to be optimal. If the initial inclination of the baton is below some critical angle (e.g. in a position analogous to a deep squat), the optimal control is to apply maximum torque until lift-off. On the other hand, if the initial inclination of the baton is above that critical angle (e.g. in a position analogous to standing), the optimal control is to first move the baton downward (i.e. countermove) before applying maximum torque until lift-off. With this analytical result in mind, together with our own analytical and experimental findings, we now propose that humans perform countermovements not so much to store and re-utilize elastic strain energy during jumping, but rather to increase ground contact time during the propulsion phase of the jump. By allowing muscles to shorten and do positive work on the skeleton for a longer period of time, we hypothesize that the preparatory, countermovement phase of jumping is analogous to the first phase of the optimal control solution for propelling a baton to a maximum vertical distance (Levine et al., 1983). Since jump height is determined by the magnitude of the vertical acceleration of the whole-body center of mass and by total ground contact time, both of which determine the area under the vertical ground reaction force, the vertical acceleration of the center ofmass, and more specifically its variation with body position (i.e. vertical displacement), appears, at least intuitively. to be an important quantity. Given that muscles are approximately isometric at the lowest position of a countermovement, we used our model to determine the envelope of the maximum vertical acceleration of the center of mass for various positions of the body ranging from a deep squat to standing. First, we applied peak. isometric, extensor torques at the ankle, knee, and hip, and found that the vertical acceleration of the center of mass reaches a maximum at a body position slightly lower than standing, but then decreases as the center of mass is lowered further (Fig. 14(a). peak). This result is unaltered by either the force-length property of muscles or the musculoskeletal geometry of the model (i.e. moment arms). By applying maximum, isometric, extensor torques to 1427 each joint, taking into account the appropriate torque-angle relationships (Pandy et al., 1990), we found that the vertical acceleration of the center of mass once again reaches a maximum just below standing, and then decreases for lower body positions (Fig. 14(a), max). Because lower body positions do not bring increasing vertical accelerations of the center of mass of the body, we conclude that humans do not countermove to increase the vertical acceleration of their center of mass during jumping. To complete the above picture, we have included the effect of muscle’s force-velocity property by plotting the variation of the vertical acceleration of the center of mass with body position during a CMJ and an SJ (Fig. 14(a), CMJ and SJ). Because the magnitude of the vertical acceleration remains approximately at or below the levels generated by both the maximum, isometric joint torques and the peak joint torques (Fig. 14(a), compare CMJ and SJ with peak and max at 0.30-0.45 cm below standing), we contend that jumping performance is determined by the time over which muscles accelerate and deliver energy to the body segments and not by the magnitude of the vertical acceleration induced by muscles during the jump. In this respect, a maximum-height CMJ is analogous to the optimal control solution for propelling a baton to a maximum vertical distance. In either case, the requirement of a preparatory countermovement is to increase the propulsion time of’ the jump (i.e. the time over which maximum torque is exerted at any joint). To support the above contention, we computed the optimal controls for a maximum-height CMJ and a maximumheight SJ when the model begins from a position in which its center of mass is just 1Ocm below standing. From this intermediate position, even though the model generates much larger vertical accelerations during the SJ than it does during the CMJ (Fig. 14(b). 25 ms-* for the SJ compared to only 15 msV2 for the CMJ), the CMJ is significantly higher (64 cm for the CMJ compared to only 47 cm for the SJ). This, together with the fact that the propulsion phase of the CMJ lasts 1.1 s compared to only 0.51 s for the SJ [not seen in Fig. 14(b)], supports our contention that higher jumps are generated by countermoving to lower body positions in order to increase the propulsion time of the jump.