Download storage and utilization of elastic strain energy

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.
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Anderson,
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ing. J. Biomechanics 19, 899-906.
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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.