JOURNALOF NEUROPHYSIOLOGY
Vol. 7 1, No. 4, April 1994. Printed in U.S.A.
A Control Strategy for the Execution of Explosive Movements
From Varying Starting Positions
ARTHUR
J. VAN
SOEST, MAARTEN
F. BOBBERT,
AND
GERRIT
JAN VAN
INGEN
SCHENAU
Faculty of Human MovementSciences,Vrije Universiteit,NL 1081 BT Amsterdam, The Netherlands
exerted on the ground be properly controlled (van Ingen
Schenau et al. 1992; Jacobs and van Ingen Schenau 1992).
1. Humans can execute explosive movements such asjumping
Obviously, successful execution of extremely fast moveand hitting an object irrespective of the starting position from
skeletal system subwhich these movements have to be initiated; in fact, variability of ments of an inverted pendulum-like
kinematic parameters has been shown to decreasein the course of jected to strong dynamic coupling between segments poses
the movement.
extreme demands on the control system. Thus it is rather
2. We address the question of whether it is necessaryto adapt surprising that quite successful performance of these tasks
the stimulation pattern of the muscles to such variations in start- is seen in virtually all humans. In expert performers kineing position or whether the stabilizing effect of intrinsic muscle matic patterns have been shown to be highly consistent for
properties is such that one single muscle stimulation pattern might
tasks such as vertical jumping (Bobbert and van Ingen
be used for a wide range of starting positions.
Schenau 1988), the forehand drive in table tennis
3. Specifically, we address this question for maximum-height
human vertical squat jumping, using an approach based on mathe- (Bootsma and van Wieringen 1990), and cricket batting
matical modeling and computer simulation. The stimulation pat- ( McCleod 1987). Perhaps the most surprising finding is
tern of the muscles is the input of the model and the resulting that execution does not seem to be hampered when the
movement has to be started from widely different starting
movement is the output.
4. The optimal stimulation pattern for a starting position in the positions. For example, the adequacy of performance of
middle of the range of starting positions considered does not lead squat jumps (i.e., maximally high vertical jumps starting
to adequate performance for other starting positions in that range. from a static squatted position) does not depend to any
5. However, a muscle stimulation pattern can be found that degree on the initial position (M. F. Bobbert, K. G. M.
does result in close to optimal achievement for a wide range of Gerritsen, M. C. A. Litjens, and A. J. van Soest, unpubstarting positions. This muscle stimulation pattern, which is not
optimal for any specific starting position, may be considered as lished data). A similar conclusion was drawn by Bootsma
and van Wieringen ( 1990) in their study of the table tennis
“control that works” as opposed to “optimal control.”
6. The latter muscle stimulation pattern also leads to adequate forehand drive. These findings imply that variability of imbehavior for “new” starting positions both within and outside the portant movement parameters decreases as the instant of
time on which achievement depends (i.e., takeoff in jumprange considered.
ing; ball contact in table tennis) is approached.
The combination of high task demands and observed sucINTRODUCTION
cessful performance leads to the general question of which
control strategy is employed by performers of such tasks. A
In animals, skillful execution of explosive movements
such as jumping, sprint running, throwing, kicking, or hit- communis opinio exists with respect to the fact that the
ting an object is of paramount importance both in fleeing control of explosive movements is to a certain extent preprogrammed (Schmidt 1982). However, the opinions conbehavior and in prey catching. In humans these movements
cerning two more specific questions are less unequivocal.
are prominent primarily in sport situations. Roughly
stated, the purpose of these tasks is to maximize the final The first of these concerns the way in which disturbances
that take place during the unfolding of the movement are
velocity of the end point of a skeletal chain. Joint angular
velocities reach values as high as 1,000” /s in human verti- counteracted. The second question is how the observed
cal jumping (Bobbert and van Ingen Schenau 1988)) which large variations in starting position are dealt with.
Adaptation of the motor program on the basis of propriois one of the slower explosive movements. The strong dyceptive feedback is the first mechanism that springs to mind
namic coupling between the segments that results from
in relation to the first question: it is imaginable that the
such velocities complicates the control of these movements
muscle stimulation pattern is
significantly (Hollerbach and Flash 1982). A related char- default preprogrammed
adapted when information on a disturbance of the moveacteristic is the short duration of the movement; movement
times for these tasks in humans vary from -40 ms for Mu- ment becomes available on the basis of proprioceptive feedhammed Ali’s left jab (Schmidt 1982) to -300 ms for the back.. However, it is well established that in the lower expushoff phase in vertical jumping (Bobbert and van Ingen tremity the fastest neural feedback loop, i.e., the myotatic
Schenau 1988). Finally, the skeletal system is mechanically
reflex, involves a latency of -45 ms between stimulus and
analogous to an inverted pendulum in cases where the end first change in electromyogram (EMG) and that the gain of
point moves against gravity. Vertical jumping is a clear ex- this reflex at high velocities is low (Gottlieb and Agarwal
1979). In the upper extremity this latency is somewhat
ample in this respect. Preventing disintegration of the movement in such cases requires that the direction of the force shorter, i.e., -30 ms (Gielen and Houk 1984). FurtherSUMMARY
1390
AND
CONCLUSIONS
0022-3077/94
$3.00 Copyright 0 1994 The American Physiological Society
CONTROL
OF EXPLOSIVE
more, although a certain degree of interjoint coordination
exists with respect to this reflex (Lacquaniti and Soechting
1986; Smeets and Erkelens 199 1), it is unlikely that these
effects can be modulated to such an extent, that the timevarying dynamic coupling between segments can be counteracted adequately. In contrast, the “functional stretch reflex” (Melville Jones and Watt 197 1) can be modulated to
a large extent; however, this reflex is characterized by a
much longer latency, typically - 120 ms. Also, it must be
considered that muscle force lags behind EMG by -5O100 ms (Vos et al. 1990); these forces determine acceleration, which has to be integrated over time once or twice to
obtain changes in velocity/position,
which again takes
some time. Because all these processes occur serially all latencies involved have to be added, leading to a total latency
of - lOO- 150 ms at the minimum for a feedback loop that
can be modulated only to a limited extent. Comparing
these latencies to the movement times involved it is inevitable to conclude that adaptation of the stimulation pattern
of muscles on the basis of neural feedback cannot be effective in counteracting disturbances occurring during explosive movements.
A second mechanism that has been suggested to reduce
the effects of disturbances occurring during movement concerns the stabilizing effect of muscle properties. The contribution of the force-length-velocity relation of muscle to the
counteracting of externally imposed disturbances has been
known for quite some time (e.g., Grillner 1972). In fact, the
intrinsic stiffness resulting from the force-length relationship of muscles below optimum length is at the heart of the
equilibrium point theory of Bizzi et al. (e.g., Bizzi et al.
1992). Although this intrinsic stiffness is smaller than that
resulting from the myotatic reflex (Carter et al. 1990; Nichols and Houk 1976)) it has the advantage of zero time delay
(Hogan 1990; McMahon 1984) : an adaptation of muscle
force occurs instantaneously when muscle length and/ or
velocity are disturbed. This zero time delay might be an
especially important property in fast movements, because
in these movements the role of neural feedback is limited.
Recently we investigated the potential contribution of the
force-length-velocity
relation of muscle in counteracting
disturbances occurring during movement in human squat
jumping (van Soest and Bobbert 1993). Using a modeling
and simulation approach it was shown that these muscle
properties in fact reduce the effect of disturbances to such a
degree that an adaptation of the muscle stimulation pattern
is no longer needed. Because of muscle properties the control system is relieved of the burden of adapting the muscle
stimulation pattern in case of disturbances.
Having identified the mechanism most likely to be responsible for handling of disturbances occurring during
movement, we now turn to the question of how the observed large variations in starting position are dealt with.
Obviously these variations may be considered as disturbances of a “standard” starting position. As such, this question may be considered as a special case of the question
discussed above. However, explosive movements are typically started from a static position that was maintained for
quite some time. Consequently, neural feedback of information concerning this position is definitely possible. Thus
the question of how variations in initial position are han-
MOVEMENTS
1391
dled is usually replaced by the more specific question of
how the control system succeeds in generating an adequate
muscle stimulation pattern on the basis of “knowledge” of
the initial position. Several propositions have been made
with respect to this more specific question. On the one
hand, it has been suggested that an adequate “specific motor program” is either retrieved from storage directly or
created by setting a number of parameters in a stored “general motor program” (Schmidt 1982). On the other hand,
it has been suggested (Cooke and Diggles 1984; van der
Meulen et al. 1990) that adequate control signals are generated “on-line” on the basis of an efference copy (von Holst
and Mittelstaedt 1950). A characteristic shared by all these
suggestions is that they place heavy demands on the capabilities of the control system.
In the present study we attempt to formulate a less involved control strategy for handling large variations in
starting position. The alternative strategy proposed is based
on the finding that muscle properties reduce the effect of
disturbances adequately. Generalizing this finding, we hypothesize that a single muscle stimulation pattern exists
that yields adequate performance for a wide range of starting positions. Direct experimental confirmation of any of
the control strategy hypotheses mentioned is difficult. As a
start, we feel it is justified to test our hypothesis using a
modeling and simulation approach. The results of this test
are described in this study. After showing that the salient
features of the “real” system are adequately represented in
the model, it will be shown that it is indeed possible to find a
single muscle stimulation pattern that results in successful
performance of human squat jumps initiated from a wide
range of static starting positions. Once more, we stress that
a control strategy based on using such a stimulation pattern
would imply an enormous simplification of the control
problem associated with handling variations in starting position when compared with the specific motor program and
“efference copy” hypotheses. After showing in this study
that a simple control strategy along the lines of our hypothesis might be used, the next step is to investigate whether
such a strategy is actually used in reality. As a first step in
this direction we will show that available experimental data
can be explained by the hypothesis presented in this study.
METHODS
Outline of the study
The central question addressed in this study is whether successful performance of squat jumps initiated from widely varying
static starting positions might be achieved by using a single suitably selected muscle stimulation pattern. This question is investigated using a modeling and simulation approach. After supplying
a concise description of the task studied we will describe the model
of the musculoskeletal system. In the simulations a simple representation of the activation of alpha-motoneurons as a function of
time (henceforth called STIM pattern) and the initial position
constitute the input to the model; the resulting movement constitutes the output of the model. To evaluate whether the selected
STIM pattern (see below) yields successful performance when
used in a specific starting position, the resulting jump height must
be compared against the maximal jump height for that starting
position. Thus for any starting position the optimal STIM pattern
and the corresponding optimal movement and jump height must
1392
A. J.
VAN
SOEST,
M.
F. BOBBERT,
be used as a reference. The details of the numerical procedure
involved in the optimization of the STIM pattern will be described.
Although a simulation approach is used in this study, it is the
real system that we are ultimately interested in. Therefore it must
be shown that the salient features of the real system are adequately
represented in the model. To this aim we made a comparison
between the performance of well-trained human subjects and optimal performance of the model. The assumptions underlying this
comparison are that (I) the optimization criterion used in the
simulations, i.e., height reached by the body’s center of mass, is the
one also used by our subjects and (2) that well-trained subjects
perform more or lessoptimally. A short description of the experimental procedures is given.
The general approach used in the simulation experiments
aimed at finding one STIM pattern that yields successfulperformance for a range of starting positions can be summarized as
follows: 1) select a STIM pattern; 2) apply this STIM pattern to a
number of static starting positions; and 3) compare the resulting
movement patterns and jump heights to the optimal movement
patterns and jump heights pertaining to these starting positions.
Two simulation experiments were conducted, differing in the
STIM patterns used. The STIM pattern used in simulation experiment I was an obvious choice: the optimal STIM pattern pertaining to the standard (middle) starting position. In simulation experiment II no STIM pattern was selected a priori. Rather, we
attempted to find a STIM pattern that results in successfulperformance for all starting positions considered. In addition, we investigated how successfulperformance was when the resulting STIM
pattern was applied to “new” starting positions, i.e., starting positions that were not used in the derivation of this STIM pattern.
Descriptionof the task
The type of human vertical jumping studied is the maximumheight squat jump, i.e., a jump starting from a static squatted
position aimed at reaching a maximal height of the body’s center
of mass. Becausejump height can be calculated from the position
and velocity at takeoff, only the pushoff phase, which starts when
the body’s center of massstarts moving upward and ends when the
feet lose contact with the ground, is considered. Successof jumps
is not strictly defined in this study. Becausethe aim of the task is to
jump as high as possible,jump height is the most important determinant of success;however, the movement pattern and specifically the position of the segments and the direction of the velocity
of the body’s center of mass at takeoff are taken into account as
well.
Model of the musculoskeletalsystem
The model used in our simulations of human vertical jumping
has been extensively described elsewhere (van Soestet al. 1993). It
consists of a submodel describing the skeletal structure and a submodel describing the behavior of the muscles.
The skeletal submodel is two-dimensional and consists of four
rigid segments representing feet, lower legs, upper legs, and headarms-trunk. These segments are connected in frictionless hinge
joints representing hip, knee, and ankle joints. At the distal end of
the foot segment the skeletal model is connected to the rigid
ground by a fourth hinge joint that can be considered as a representation of the metatarsophalangeal joint. Possibleground contact at
the heel is assumed to be elastic. Acceleration-determining forces
are the gravitational forces, the force at the heel in case of ground
contact, and the moments acting at the joints that represent the
net action of the muscles. The dynamic equations of motion of the
skeletal model were derived using SPACAR, a software subroutine
package developed at Delft University of Technology (van Soestet
al. 1992; van der Werff 1977). These equations allow calculation
AND
G. J.
VAN
INGEN
SCHENAU
of the acceleration of the skeletal model as a function of position,
velocity, and the forces mentioned above. Segment lengths for the
skeletal model as well as initial position were directly based on the
mean values of the experimental subjects. The initial position
based on experimental data will be referred to as the standard
initial position. Segment masses, locations of segment mass
centers, and segmental moments of inertia were estimated from
segment lengths and body mass of our subjects, in conjunction
with cadaver data reported by Clauser et al. ( 1969).
The muscular submodel consists of six “muscles” representing
the major muscle groups that contribute to extension of the lower
extremity, i.e., gluteal muscles, hamstrings, vasti, rectus femoris,
soleus, and gastrocnemius. A Hill-type muscle model was used to
represent these muscles. It consists of a contractile element, a series elastic element, and a parallel elastic element. Behavior of the
elastic elements is governed by nonlinear force-length relationships. Behavior of the contractile element is more complex: contractile element contraction velocity depends on active state, contractile element length, and force. Force is directly related to the
length of the serieselastic element. This length can be calculated at
any instant from the position of the skeleton and contractile element length (which are used as state variables) because muscletendon complex length is directly related to position of the skeleton. Active state is related to muscle stimulation STIM, the independent neural input of the model, by first-order dynamics as
described by Hatze ( 198 1). STIM ranges between 0 and 1 and is a
one-dimensional representation of the effects of recruitment and
firing rate of alpha-motoneurons.
Wherever possible, parameter values for the muscles were derived on the basis of morphometric data. Most importantly, contractile element optimal lengths were based on sarcomere numbers; moment arms of the muscles were based on muscle length
versusjoint angle measurements; relative values for maximal isometric forces were based on muscle cross-sectionalareas. All these
measurements were made on the same group of cadavers. Sarcomere numbers and moment arms were scaled in proportion to
segment lengths; absolute values of muscle forces were set in such
a way that realistic maximal isometric moments were obtained.
The force-velocity relation was based on Hill’s classicalequation
(Hill, 1938)
where FcEis contractile element force, FyIAxis maximal isometric
force, V& is contractile element contraction velocity, and LCE(OPT)
is contractile element optimal length. The dimensionless parameters dFnfi*x and WCE(OPT)were set to 0.41 and 5.2, respectively,
for all muscles. The stretch of the serieselastic element at maximal
isometric force was set at 0.04 times its slack length for all muscles.
A detailed discussion of the structure of the muscle model and the
strategy followed in the estimation of parameter values has been
presented elsewhere (van Soest and Bobbert 1993; van Soestet al.
1993).
In total, the model is mathematically described by a set of 20
coupled nonlinear first-order ordinary differential equations.
Given the initial state and given the independent control signals
(i.e., STIM) as a function of time, the resulting movement can be
calculated by numerical integration. A variable-order variablestep sizeAdams-Bashford predictor Adams-Moulton corrector integration algorithm was used (Shampine and Gordon 1975).
Optimization of STIM pattern
The problem is to find the STIM( t) pattern that leads to a maximal vertical jump, i.e., to a maximal height of the body’s center of
mass. This dynamic optimization problem was studied in its full
complexity by Pandy et al. ( 1990). Partly on the basis of their
results a more restricted form of dynamic optimization is used in
CONTROL
OF EXPLOSIVE
this study. Specifically, the following restrictions were imposed on
STIM: first, the initial STIM level was set in such a way that the
static squatting starting position was maintained. Second, STIM
was allowed to take on either this initial value or the maximal
value of 1.0. Third, STIM was allowed to switch to its maximal
value just once and thereafter remained maximal until takeoff.
Under these restrictions, STIM ( t ) of each of the six muscle groups
is described by a single parameter: the instant at which STIM
switches from initial value to maximal value. The optimization
problem is thus reduced to finding the combination of six switching times that results in maximal jump height. Although this is still
a computationally demanding problem, it can be solved using
standard algorithms. NAG subroutine E04UCF, a sequential quadratic programming algorithm, was used (NAG Fortran Library
Manual Mark 13, Numeric Algorithms Group, Oxford, UK). In
the following, the optimal STIM pattern pertaining to the initial
position derived from experimental data will be referred to as the
standard STIM pattern and similarly the resulting movement will
be referred to as the standard movement.
Severe constraints were imposed on STIM in this study to keep
calculation time manageable and to ensure that a well-defined
optimum could be found. To investigate the extent to which these
constraints influenced performance, optimizations were performed where each muscle was given more “freedom:” each was
allowed to switch on two times and to be deactivated in between.
Although this enhanced jump height, the enhancement was only
~2 mm. Furthermore, numerous local optima appeared to exist.
It is concluded that by reducing the dimension of the control space
to 6, as is done in this study, the optimization problem becomes
more well-behaved and the reduction in jump height is small.
Experimental procedures
Six elite male volleyball players performed a number of jumps
starting from a freely chosen static squatting position. Subjects
were instructed to jump as high as possible, to make no countermovement before pushoff, and to keep their hands on their backs.
Trials in which these instructions were violated were discarded.
The highest successful jump of each subject was selected for
analysis.
During jumping, kinematic data were gathered using a lOO-Hz
VICON video registration and analysis system and ground reaction force was measured using a force platform (Kistler 928 1B) .
These data were used for an inverse dynamic analysis using standard procedures ( Bobbert et al. 1993 ) . Surface EMG was recorded
simultaneously using a Biomess 80 telemetric system. After amplification and analog band-pass filtering (20-200Hz), EMG data
were sampled at 1,000 Hz and stored in digital form.
Simulation
experiment I: locally optimal STIMpattern
As stated above, the hypothesis tested here is simply that the
optimal STIM pattern pertaining to a “preferred” static starting
position is always used and that this results in successful performance for a wide range of starting positions around the preferred
one. To test this hypothesis the following steps were taken. First it
was decided to treat the experimentally derived standard starting
position asthe preferred starting position and to treat the standard
STIM pattern correspondingly. Next two additional starting positions were defined. All three starting positions are shown in Fig. 1
(positions 1, 3, and 5). In all three, the horizontal position of the
body’s center of mass and the point of ground contact at the metatarsophalangeal joint are equal. The two additional starting positions differ from the standard one in the vertical position of the
body’s center of mass by amounts of to.1 m. As for the standard
starting position, initial state of the muscles was calculated from
1393
MOVEMENTS
1
2
3
4
5
6
FIG. 1. Static initial positions used in the simulation experiments. Dot:
position of body’s center of mass. Position 3 was derived from experimental data. See text for details.
the demand of static equilibrium. Next the standard STIM pattern
was applied to the additional two starting positions. As explained
earlier, for comparison the optimal movement pattern and optimal jump height for the additional starting positions had to be
obtained as well. This was done using the optimization approach
described above. Finally, for each of the additional starting positions, the performance resulting from application of the standard
STIM pattern was compared with the optimal performance for
that starting position.
Simulation
experiment II: globally optimal STIM pattern
As indicated earlier, no STIM pattern is selected a priori in this
simulation experiment. Rather, the question asked here is whether
a single STIM pattern can be found that resultsin successfulperformance for a number of starting positions. Note that if such a
pattern can be found it is not likely to be optimal for any specific
starting position! Despite the stringent constraints imposed on
STIM in this study, the number of possible STIM patterns is still
extremely large. As a consequence it is not possible to find the
STIM pattern we are looking for by checking all possible STIM
patterns. Instead the following approach was adopted. First, it was
decided to use the three starting positions used in simulation experiment I (Fig. 1, positions 1,3, and 5 ) asthe “learning set” in the
determination of the globally optimal STIM pattern. Next we decided to consider averagejump height for these three starting positions as the measure of successof any specific STIM pattern. In
other words, the best STIM pattern was defined to be the one that
yields the highest average jump height when applied to all three
starting positions. Obviously, finding this STIM pattern constitutes an optimization problem. In fact, this optimization problem
is similar to those encountered earlier, and consequently the numerical optimization method described earlier was fully applicable. Application of this optimization procedure resulted in a STIM
pattern that will be called the “globally optimal STIM pattern.”
Next, to evaluate just how successfulthe performance resulting
from application of this globally optimal STIM pattern to all three
starting positions was, this performance was compared with optimal performance for these three static starting positions, as obtained earlier. Finally we investigated whether the globally optimal STIM pattern resulted in successful performance when applied to starting positions that were not included in the learning
set. To this aim, three additional starting positions were defined,
two of which fell within the range covered by the learning set and
one of which fell outside that range. The globally optimal STIM
pattern was applied to these three additional starting positions and
resulting performance was compared to the relevant optimal performance. These additional initial positions are shown in Fig. 1,
positions 2, 4, and 6. Positions 2 and 4 are midway (in terms of
vertical position of the body’s center of mass) between the three
A. J.
1394
VAN
SOEST, M. F. BOBBERT,
AND G. J.
VAN
INGEN SCHENAU
A
GLU
HAM
VAS
REC
SOL
GAS
T = 0.0
T = -0.310
-0
1
35
-0.3
1
I
I
I
I
-0.25
-0.2
-0.15
-0.1
-0.05
0
TIME [s]
.
cccc
T = -0.370
I
c
T = 0.0
,
-0.9
I
-0.8
1
-0.7
1
-0.6
1
-0.5
1
-0.4
,
-0.3
I
-0.2
1
-0.1
0
TIME [s]
2. A : stick-figure representation of the pushoff movement of the model resulting in maximal jump height. Velocity
of the body’s center of mass is represented by a vector that originates from the body’s center of mass. B: stick-figure
representation of the mean movement of well-trained experimental subjects performing a maximal squat jump ( yt = 6). C:
muscle stimulation as a function of time for gluteal muscles (GLU), hamstrings ( HAM), vasti (VAS), rectus femoris
(REC), soleus ( SOL), and gastrocnemius (GAS) muscles. Drawn lines: period of maximal activation of alpha-motoneurons
as a function of time (STIM). Top line of each pair concerns the optimal STIM pattern for starting position 3; bottom line
concerns the globally optimal STIM pattern. D: typical example of surface electromyogram (EMG) patterns and vertical
ground reaction force (top tracing) obtained during pushoff. ST, semitendinosus; BF, biceps femoris, caput longum. Vertical
line: start of the pushoff phase. Note the difference in scaling of the time axes between C and D. In all figures, time is
expressed relative to takeoff.
FIG.
starting positions used earlier (i.e., positions 1, 3, and 5 in Fig. 1).
Vertical position of the body’s center of mass in position 6 is 5 cm
higher than in position 5.
RESULTS
Comparisonof experimentaldata and simulation data
Before showing results pertaining to the main question of
this study it must be shown that the salient features of the
real system are adequately represented in the model used.
To this aim, simulation data pertaining to the optimal (i.e.,
maximum-height)
jump are compared with experimental
data. In Fig. 2, A and B, stick-figure representations of the
movement are given both for our experimental subjects
( Fig. 2 B) and for the optimal simulation data ( Fig. 24.
Note that the initial position for the model was set to equal
that of our experimental data; the correspondence in this
respect is imposed. Comparing these stick diagrams a number of general observations can be made. First, the general
kinematic patterns are similar: a continuous extension occurs at all joints except at the ankle joint in the first part of
pushoff. Second, a near-perfect alignment of body’s center
of mass with the point of ground contact is observed. Third,
CONTROL
TABLE
1.
Experiment
Simulation
OF EXPLOSIVE
MOVEMENTS
1395
Comparison of experimental and simulation data of maximum-height squat jumps
H, m
Y cm9 m
@a,rad
0.447
0.098
0.048
0.52
0.60
0.390
+k,
rad
0.12
0.40
(bh,
l-ad
0.09
0.5 1
Ycm,
m/s
2.67
2.59
dir,, rad
w, J
0.0525
0.0430
612
657
All jumps are from starting position 3 (see Fig. 1). H, jump height relative to upright standing; y,,, height of body’s center of mass at takeoff, relative to
upright standing; &, &, +h, ankle, knee, hip, angle at takeoff, relative to full extension; pcrn,vertical velocity of body’s center of mass at take-off; dir,,
absolute angle of the velocity vector at takeoff, relative to the vertical; W, total mechanical work done by the muscles. Experimental data are averaged
over subjects (n = 6). See text for details.
the velocity of the body’s center of mass at takeoff is almost
perfectly vertical.
In Fig. 2C (top line of each pair) the optimal STIM pattern is presented. In Fig. 2D, surface EMG tracings for a
maximum-height squat jump of a typical subject are shown
together with the vertical ground reaction force. Taking
surface EMG as a measure of the muscle activation level,
two observations can be made when comparing Fig. 2, C
and D. First, the EMG data support the assumption that,
once activated, muscles remain active until takeoff, although maximal activity is not maintained in all muscles.
Second, the notion of a proximal-distal sequence in muscle
activation (e.g., Bobber-t and van Ingen Schenau 1988) is
confirmed by both the EMG data and the optimal STIM
pattern.
To allow a more quantitative comparison between experiment and simulation, a number of parameters related to
achievement in vertical jumping are presented in Table 1.
The first thing to note from this table is that jump height for
the model is 13% lower than that observed experimentally.
As analyzed in detail in van Soest et al. ( 1993 ), this is because of the fact that the upper body is represented by a
single rigid segment in the model, as a result of which trunk
extension cannot contribute to jump height. This trunk extension by itself results in raising the body’s center of mass
at takeoff; additionally, the inertial forces associated with
trunk extension postpone takeoff, as a result of which the
joints of the lower extremities are closer to full extension in
reality than in simulation (see Table 1). Thus the difference
in jump height is largely caused by a higher position of the
body’s center of mass at takeoff; the difference in vertical
velocity of the body’s center of mass is relatively small. For
both experiment and simulation, the direction of the velocity vector at takeoff is within 3O of the vertical; thus very
little work is invested in horizontal kinetic energy. Finally,
the mechanical work (i.e., sum over all joints of the integrals of net joint moments over joint angle) done by the
muscles during pushoff is compared. Given the difference
in jump height, it is surprising that the total work is higher
in simulation. The unexpectedly low value of the experimentally observed muscle work is, again, due to trunk deformation: in movement registration and inverse dynamic
analysis the trunk was assumed to be rigid. Thus in the
experimental value work done by the upper body muscles is
neglected, which explains the relatively low work value calculated from the experimental data. Additional data on this
comparison of experiment and simulation have been presented in van Soest et al. ( 1993). It is concluded that although simplifications in the musculoskeletal model result
in certain differences the degree of correspondence between
experimental and simulation data is such that the model
can be used with confidence to investigate the question addressed in this study.
Simulation
experiment I: locally optimal STIMpattern
In this simulation experiment we tested the hypothesis
that the optimal STIM pattern pertaining to a specific starting position is always used and that this strategy results in
successful performance for a wide range of starting positions around that specific one. To test this hypothesis we
applied the optimal STIM pattern pertaining to the experimentally derived standard starting position (standard
STIM) to two additional starting positions (Fig. 1, positions 1 and 5). This resulted in movements that are presented in the form of stick diagrams in Fig. 3, D and E,
along with corresponding jump heights relative to upright
standing. For comparison, the optimal performance pertaining to these starting positions is presented in Fig. 3, A
and C. In addition, values of a number of parameters related to achievement in vertical jumping are presented in
Table 2. Both for the optimal jumps and for the jumps
resulting from applying standard STIM to starting positions
1, 3, and 5, average values are presented in Table 2.
Comparison of Fig. 3, A and Cand D and E indicates that
application of the standard STIM pattern to widely varying
starting positions results in movements that differ considerably from the optimal movements for these starting positions. For example, in starting position 1 hyperextension of
the hip is present at takeoff, which clearly does not occur in
either the optimal jump for that starting position or in reality. Even more importantly, jump height is seriously affected ( see also Table 2 ) : averaged over starting positions 1,
3, and 5 jump height is 0.115 m less, which in a relative
sense amounts to as much as 30%. This deterioration of
performance is for the major part because of a lower vertical velocity at takeoff and to a lesser extent because of a
lower position of the body’s center of mass at takeoff. In
turn, this lower velocity is caused by a lower amount of
energy delivered by the muscles during pushoff: whereas
the averaged value for the work done equals 652 J for the
optimal performance, it equals 558 J on average in cases
where standard STIM is applied to each of the starting positions 1, 3, and 5. Although the direction of velocity at takeoff diverges more from the vertical when applying standard
STIM than when applying optimal STIM (0.0992 vs.
0.0405 rad), the importance of this parameter in terms of
work wasted in the form of horizontal kinetic energy is
small.
In conclusion, the hypothesis that application of standard STIM to a wide range of starting positions results in
A. J.
T = -0.356
A
VAN
SOEST, M. F. BOBBERT,
T = -0.310
AND G. J.
B
VAN
INGEN
SCHENAU
T = -0.256
H = 0.215
H = 0.208
T = -0.386
T = -0.256
D
H = 0.352
H = 0.423
T = -0.374
F
C
T = -0.306
T = -0.262
U
H
FIG. 3. Stick-figure representation of pushoff movements obtained in simulation experiments I and II concerning starting positions 1 (A, D, and F), 3 (B and G), and 5 (C, E, and H). A-C: optimal movement for these starting positions, used as
reference. D and E: movements obtained in simulation experiment I. F-H: movements obtained in simulation experiment
II. H: resulting jump height. For further details see legend to Fig. 2 and main text.
close to optimal achievement was refuted by the results of
this simulation experiment.
Simulation experimentII: globally optimal STIMpattern
In this simulation experiment we attempted to find one
single STIM pattern that results in successful performance
for a number of static starting positions. Second we investigated whether this STIM pattern results in successful performance for new starting positions. To find the globally optimal STIM pattern we performed optimization with respect
to average jump height over the initial positions 1,3, and 5
as indicated in Fig. 1. The resulting STIM pattern is depicted in Fig. 2C, together with the optimal STIM pattern
for starting position 3. For the gluteal muscles, the hamstrings, and the vasti group, the time of onset of maximal
stimulation differs by < 10 ms. Rectus femoris is activated
34 ms earlier in the globally optimal STIM pattern. The
most significant difference concerns the ankle plantar flexors soleus and gastrocnemius. In both cases they are the last
activated muscle group. In the optimal STIM pattern, the
biarticular gastrocnemius was activated 84 ms after the
monoarticular soleus; in the globally optimal STIM pattern
this sequence was reversed. This reversal can be speculated
to have a relatively small effect on the (plantar flexing)
ankle joint moment; the main consequence is that a direct
coupling between ankle and knee joints is created earlier in
the pushoff. For an extensive discussion of the relation between subtle changes in STIM and the resulting movement
the reader is referred to Bobbert and van Soest ( 1994).
The movement resulting from applying the resulting globally optimal STIM pattern to all three initial positions is
presented in Fig. 3, F-H, along with corresponding jump
heights and pushoff phase durations. Note that application
CONTROL
TABLE
2.
OF EXPLOSIVE
MOVEMENTS
1397
Values of numerical parameters related to successof simulated vertical jumps
Optimal STIM
Avg-opt STIM
Standard STIM
K m
Y cm7 l-n
0.386
0.365
0.27 1
0.048
0.045
0.017
&I,
rad
0.61
0.57
0.93
hc,
rad
0.39
0.39
0.14
4h,
I-ad
0.47
0.47
0.13
iQcrn3
m/s
2.57
2.49
2.21
dir,,
rad
0.0405
0.0343
0.0992
w J
652
638
558
Val ues are averaged over starting positions 1,3 and 5 (see Fig. 1). Optimal STIM, optimal jumps for each starting position; Standard STIM, jumps that
result from applying the optimal STIM for starting position 3; Avg-opt STIM, jumps that result from applying the single globally optimal STIM pattern.
For other abbreviations see Table 1.
of the same STIM pattern to different starting positions
leads to pushoff movements with durations varying between 0.262 and 0.374 s. Similarly, as shown in Fig. 4, the
net joint moments that result from application of the same
STIM pattern to different starting positions vary largely.
However, from Fig. 5 it is seen that application of this single
STIM pattern results in decreasing variability in joint angles as the instant of takeoff is approached. The decreasing
variability in these joint angle-time plots can be quantified
by looking at their SDS. Specifically, the SD of the joint
angles for starting positions 1, 3, and 5, summed over hip,
knee, and ankle joints, decreased from 0.66 radians at the
start of the pushoff phase to 0.49 radians at takeoff. Thus in
a relative sense this SD decreases by 26% during pushoff.
Numerical values of parameters related to achievement
are presented in Table 2, again averaged over starting positions 1, 3, and 5. In comparison with the application of
standard STIM to each of these starting positions, results
are much closer to the optimal values here. Jump height
resulting from application of this single STIM pattern to the
three starting positions is on average only 0.02 m less than
optimal, which amounts to 5% in a relative sense. Joint
angles at takeoff are virtually identical to the optimal values; as a result, the same is true for the position of the body’s
center of mass. Thus the lower jump height is due to a lower
velocity at takeoff, which in turn is due to a slightly lower
amount of work done by the muscles. Finally, the direction
of the velocity at takeoff diverges from the vertical by an
amount comparable with that found for the optimal jumps
(0.0343 vs. 0.0405 rad).
It is concluded that it is indeed possible to find one single
STIM pattern that results in close to optimal performance
for a number of static initial positions.
To test the second part of the idea, concerning generalization to new starting positions, three extra static starting positions were defined, two of which (2 and 4) fall inside the
range delimited by starting positions 1, 3, and 5 and one of
which (6) falls outside that range (see Fig. 1). Note that
starting positions 2,4, and 6 were not involved in the optimization procedure that yielded the globally optimal STIM
pattern. Application of the globally optimal STIM pattern
to these starting positions results in movement patterns presented in Fig. 6, D-F. For comparison optimal movement
patterns for these starting positions are shown in Fig. 6,
A-C. Numerical values of relevant parameters are presented in Table 3 for interpolated (2 and 4) and extrapolated (6) starting positions separately. For these extra starting positions jump height resulting from application of the
globally optimal STIM pattern is -0.04 m. less than optimal. As would be expected, this difference is larger than
that for the starting positions used in finding the globally
optimal STIM pattern, which was ~0.02 m (see Table 2).
Yet the difference is still 5 10% from optimal. As a consequence, the same holds true for the variables that determine
jump height, i.e., height of the body’s center of mass and
body’s center of mass vertical velocity at takeoff (see Table
3). As can be seen from Fig. 6 and Table 3, a similar trend is
also found for joint angles at takeoff and the verticality of
the jump: the differences from optimal values are small, but
larger than those for the initial positions involved in the
derivation of the globally optimal STIM pattern. Note that
the “interpolated” starting positions (2 and 4) do not result
in performance that is clearly closer to optimal than the
performance obtained when starting from position 6 that
was outside the range used in finding the globally optimal
STIM pattern. In conclusion, performance resulting from
application of the globally optimal STIM pattern to new
starting positions is 5 10% from optimal. This is true not
only for starting positions lying in the range used in the
derivation of the globally optimal STIM pattern but also for
a starting position outside that range.
DISCUSSION
In this study we generated a parsimonious explanation
for the observation that humans are able to perform explosive movement tasks from various starting positions. According to the explanation this remarkable ability might be
achieved on the basis of an equally remarkable control strategy: using one and the same muscle stimulation pattern for
all initial positions! Support for this explanation was gathered on the basis of simulation experiments. The idea that
such a strategy might work was arrived at on the basis of
simulation results described elsewhere (van Soest and Bobbert 1993) that dealt with the question how disturbances
occurring during the execution of explosive movements are
handled. In that study it was shown that intrinsic muscle
properties counteract such disturbances to such a degree
that adaptation of the muscle stimulation pattern is not
necessary. A possibility for experimental verification of this
finding lies in direct measurement of joint stiffness and
damping during movement. Such measurements were recently undertaken by Bennett et al. ( 1992) during slow,
low-force arm movements. Because stiffness and damping
during such movements are primarily determined by
neural feedback-based adaptation of muscle activation, as
opposed to by intrinsic muscle properties, the results of
Bennett et al. ( 1992) can unfortunately not be generalized
to the fast, high-force explosive movements discussed in
this study.
Two STIM patterns that possibly result in successful performance when applied to a range of starting positions were
1398
4OOb
1
.
SOEST, M. F. BOBBERT,
.
350-
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VANVAN
A. J.
AND G. J.
SCHENAU
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TIME
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4. Net joint moments versus time (relative to takeoff) for nip
(top), knee (middZe), and ankle (bottom) joints, resulting from applying
the globally optimal STIM pattern obtained in simulation experiment II to
starting positions 1 ( -),
3 ( - l ), and 5 (- - -). See text for details.
TIME [s]
5. Joint angles vs. time (relative to takeoff) for hip (top), knee
(middle), and ankle (bottom) joints, resulting from applying the globally
optimal STIM pattern obtained in simulation experiment II to starting
positions 1 ( -),
3 ( - l ), and 5 (- - -). See text for details.
investigated. These STIM patterns represent two ends of a
continuum. The first of these is the optimal STIM pattern
for a “middle” starting position. To arrive at such a pattern,
humans would have to “tune” the stored muscle stimulation pattern in such a way that it yields optimal performance when applied to a preferred starting position. Evi-
dently application of this muscle stimulation pattern to
other starting positions will result in suboptimal performance; because of the effect of muscle properties, however,
performance will be successful as long as starting positions
are sufficiently close to the preferred starting position. The
second STIM pattern is optimal with respect to average
l
FIG.
l
CONTROL
OF EXPLOSIVE
MOVEMENTS
1399
H = 0.306
j
i
T = -0.336
T = -0.250
A
H = 0.361
T = -0.202
B
C
H = 0.273
H = 0.323
1
T = -0.336
T = -0.268
D
T = -0.234
E
F
6. Stick-figure representation of pushoff movements concerning starting positions 2 (A and D), 4 (B and E), and 6
( C and F) . A-C: optimal movement for these starting positions, used as reference. D-F: movements resulting from applying
the globally optimal STIM pattern obtained in simulation experiment II to these “new” starting positions. For further details
see legend to Fig. 3 and main text.
FIG.
jump height achieved. To arrive at this pattern humans
would have to tune the stored muscle stimulation pattern in
such a way that reasonably successful performance is obtained for a wide range of starting positions. In other words,
such a strategy is aimed at establishing “control that works”
(Gottlieb et al. 1990; Hogan and Winters 1990) as opposed
to optimal control.
In simulation experiment I we found that the locally optimal STIM pattern does not result in successful performance for a wide range of starting positions. Thus this pattern is not the one we are looking for in this study. It seems
that the price that has to be paid for local optimality is that
the range of starting positions for which successful performance is obtained is relatively narrow. Although it is conceivable that expert performers are willing to narrow the
range of allowed starting positions in return for higher
achievement, such a strategy is probably not favored by
most people.
In simulation experiment II we found that the globally
TABLE
3.
optimal STIM pattern results in successful performance for
a wide range of starting positions. Thus our hypothesis was
confirmed in this experiment: a single control pattern that
works for a wide range of starting positions can indeed be
found. In simulation experiment II it was further shown
that the STIM pattern that works works for new starting
positions as well. Thus the novelty problem, traditionally
one of the weak points of motor program theories, does not
apply to the theory presented here. In conclusion, the results of simulation experiment II convincingly show that in
return for an insignificant loss ofjump height an enormous
simplification of the control problem can be obtained: the
control problem is reduced to learning, storing, and retrieving just one muscle stimulation pattern.
Having shown that the proposed control strategy works
in a simulation context, two questions must be considered:
I) would such a strategy work in reality and 2) is it actually
used in reality. An affirmative answer to the first question
can be given only if we are confident that the model ade-
Values of numerical parameters related to successof simulated vertical jumps
K
Positions 2 and 4
Optimal STIM
Avg-opt STIM
Position 6
Optimal STIM
Avg-opt STIM
l-n
Y cm9 In
(b,, rad
AC, I-ad
0.386
0.342
0.050
0.044
0.64
0.38
0.38
0.48
0.43
0.67
0.306
0.273
0.058
0.05 1
0.60
0.72
0.35
0.21
0.34
0.25
Values concern starting positions 2 and 4 (interpolation)
and 6 (extrapolation)
h,
rad
dir,, rad
w9 J
2.57
2.42
0.0463
0.0406
642
624
2.21
2.09
0.0153
0.0104
445
412
iQcm9 m/s
(see Fig. 1). For abbreviations see Table 1.
1400
A. J.
VAN
SOEST, M. F. BOBBERT,
quately represents the salient features of the real system.
This confidence was gained on the basis of a comparison of
experimental data concerning expert performance and simulation data concerning optimal model performance. In
other words it is likely that the strategy suggested would
work in reality. Thus the question remains whether it is
actually used. At present experiments are being conducted
that address this question.
A tentative theory
The theory emanating from these simulation experiments can be summarized as follows. Execution of explosive movement tasks is governed largely by open loop control. For specific groups of tasks, e.g., “explosive leg extensions,” a muscle stimulation pattern is stored in some at
present unspecified form within the CNS. Any time such a
task has to be performed the muscle stimulation pattern is
called on. Disturbances occurring during the unfolding of
the movement cannot be corrected fully on the basis of
neural feedback because of stringent time constraints. Instead such disturbances are initially counteracted by the
viscoelastic behavior resulting from the force-length-velocity relation of muscle. The stored pattern is molded by experience. Typically this results in control that works, i.e., in a
muscle stimulation pattern that yields successful performance for a wide range of starting positions. Even for starting positions that were never encountered before, reasonably successful performance is achieved.
Alternative strategies for coping with variations in starting
posit ion
As mentioned in the INTRODUCTION,
the strategy proposed in this study to cope with variations in starting position is not the only possible one that might be successful. At
least two alternative strategies can be mentioned and will be
shortly discussed here.
In the first alternative strategy, which is based on efference copy theory (von Holst and Mittelstaedt 1950), it is
assumed that the muscle stimulation pattern is formed as
the movement unfolds. The efference copy theory was formulated to explain observations like the following. When
an animal’s motions are restricted to movement of the eyes
it is obvious that identical changes in the retinal image can
result from either movement of the eye or movement of the
world. Yet from the animal’s behavior it is concluded that it
is able to distinguish between these possibilities. According
to efference copy theory, this is achieved by sending a copy
of the “motor command” to an “internal representation.”
There the eye movement that is going to result from this
command is “calculated” and this movement is “subtracted” from the movement of the retinal image. This subtraction results in the movement of the world, which solves
the animal’s problem. The central aspect of the theory is the
internal representation, in which the movement that will
result from the motor commands is calculated. Theories in
which the use of such an internal representation of the effector system is assumed are called efference copy theories.
The efference copy theory pertaining to the control of explosive movements states that the expected movement calculated in the internal representation is used in the genera-
AND G. J.
VAN
INGEN
SCHENAU
tion of the subsequent motor commands. The advantage of
the use of such an internal representation over peripheral
feedback is, of course, that delays occurring in the internal
representation are much smaller. Although the theory in its
general form is elegant, its application to the problem of
movement control requires that the internal representation
be highly accurate. For a mechanically simple system such
as the eye, existence of a sufficiently accurate representation is conceivable, and efference copy theory gives rise to
an elegant explanation. By contrast, a sufficiently accurate
representation of the motor system of the extremities would
have to be extremely complicated, much more complicated
than the simulation model used in this study, in fact. In our
view, explanations of the control of explosive movements
based on the assumption that such a complex internal representation is present in the CNS are not attractive.
The second alternative strategy is based on a priori formation of a motor program. In fact, several versions of such
a strategy can be distinguished. According to the first a single muscle stimulation pattern is stored. This stored pattern
is the optimal one for a standard starting position. Because
of muscle properties this pattern will lead to acceptable
movements for a small range of starting positions around
the standard one. To execute the task from any starting
position a relatively slow movement is initiated in the direction of the standard starting position under feedback control; as soon as the small range mentioned is reached, the
optimal muscle stimulation pattern is triggered. In fact,
such a strategy was used in an early attempt at deriving the
optimal control for a simplified model of vertical jumping
(Levine et al. 1983). However, two arguments may be
raised against such a strategy. In the first place, experimental data (Bootsma and van Wieringen 1990; M. F. Bobber-t,
K. G. M. Gerritsen, M. C. A. Litjens, and A. J. van Soest,
unpublished data) do not yield any evidence that such a
strategy is used. Second, such a strategy would result in
increases in movement time that are unacceptable in tasks
such as hitting an approaching object, in which timing
aspects are imposed externally.
According to the second version of motor program
theory, a specific muscle stimulation pattern is stored for
every starting position ( Atkeson 1989). This version is not
attractive, because it raises more questions than it answers:
the storage space required is virtually infinite; it cannot be
imagined how the correct program can be retrieved fast
enough; from the theory it is absolutely unclear how a control pattern is generated when a new starting position is
encountered (the “novelty problem”).
Finally, according to the third version of motor program
theory, a general motor program is stored in which a few
parameters are set to obtain a specific motor program (e.g.,
Carter and Shapiro 1984). In the context of the problem
considered in this study, this would imply that the specific
motor program is formed on the basis of feedback of the
initial position. This version cannot be dismissed at present; however, the relations between starting position and
values for the parameters to be set are probably not simple.
To our knowledge such relations have never been described
for any specific task.
In conclusion, alternative control strategies for dealing
with variations in initial position that are mentioned in liter-
CONTROL
OF EXPLOSIVE
ature are less parsimonious than the theory outlined in this
study. Furthermore, the theory presented in this study is
supported by concrete simulation results, which is not the
case for the theories discussed above.
Possibleneurophysiologicalimplementationof the ideas
presented
An intriguing question is how the muscle stimulation
pattern is formed in the CNS. Recent developments in the
field of neural networks provide suggestions for ways in
which time-varying patterns may be stored and retrieved
(Coolen and Gielen 1988; Reiss and Taylor 199 1) . These
authors describe relatively simple and biologically not too
implausible neural networks that are able to regenerate a
sequence of patterns without actually storing the patterns
themselves. Such a process of sequence regeneration is triggered by bringing the network into a specific initial state. It
is the strength of the connections between the neurons involved that determines what pattern is produced. As a consequence, by changing the strength of the neuronal connections involved, changes to these patterns can be made. It is
imaginable that the regenerated patterns represent the default alpha-motoneuron
activity. Alternatively these patterns might represent the control signals at a more global
level of description, in which case the details must be filled
in at a lower level. In the case of positioning tasks this global
level might well be thought of as a virtual trajectory of equilibrium points (e.g., Bizzi et al. 1992; Feldman 1986). In
explosive tasks an equilibrium point representation probably does not simplify the control very much, because an
adequate equilibrium point trajectory is expected to be
unrelated to the actual movement because of the demand
of increasing velocity; however, other global representations may be conceivable. In any case, these regenerated
patterns would, as far as possible together with feedback
signals, finally lead to activation of alpha-motoneurons.
Thus it can be imagined that the central command
“JUMP!” consists of little more than a set of signals that
triggers a network capable of reproducing the adequate
muscle stimulation pattern.
Some speculationson experimentaldata on explosive
human movements
Experiments directly addressing the theory proposed in
this study have not yet been undertaken. In fact, studies
dealing with any aspect of explosive human movements are
scarce. For some of the results of the studies that have been
performed, we will discuss to what extent they might be
explained from our theory.
An interesting set of experimental data was recently gathered by Jacobs et al. ( 1994), who analyzed the sprint start
as well as maximal vertical squat jumps of elite sprint runners. Obviously both tasks require explosive leg extension.
It was found that these subjects’ performance in the pushoff
in vertical jumping was very poor, in contrast with their
performance in the pushoff in sprint running. In the light of
the ideas presented this finding can be explained as follows.
By performing numerous sprint pushoffs sprint runners
tune the neuronal network devoted to generating the control pattern for explosive leg extensions to this specific situa-
MOVEMENTS
1401
tion. In other words, they end up with a muscle stimulation
pattern that is optimal for the sprinting situation. As was
concluded from the results of simulation experiment I, however, the price that has to be paid for obtaining optimal
performance in sprint running is a reduction in performance of related tasks that use the same muscle stimulation
pattern, such as vertical jumping.
Another group of interesting findings concerns hitting
tasks where not only is a high velocity important, but high
demands are imposed on final spatiotemporal accuracy as
well. Examples of such tasks are the attacking forehand
drive in table tennis (Bootsma and van Wieringen 1990)
and batting in cricket ( McLeod 1987). Expert performance
on all such tasks is highly consistent, meaning that intraindividual variability is small over the entire movement. However, indications exist (Bootsma and van Wieringen 1990)
that the remaining variability in fact decreases as the instant
of ball contact is approached. This finding cannot be explained on the basis of feedback, because the movement
time is well below 200 ms for this task. How then is this
reduction in variability achieved? Surprisingly, Bootsma
and van Wieringen devote no attention to this question. In
our view a reduction in variability as the instant of ball
contact is approached may result from using one single
muscle stimulation pattern for all starting positions considered. This view is based primarily on the results presented
in Fig. 5, which show for human squat jumping that when
the globally optimal STIM pattern is used variability in
joint angles decreases as the instant of takeoff is approached. The observed decrease in this case is the more
surprising because the optimization
criterion (average
jump height) was unrelated to kinematic variability. Generalizing this result to the task studied by Bootsma and van
Wieringen: a single STIM pattern is likely to exist that results in the extremely small variability of the important
movement parameters at the instant of ball contact, irrespective of starting position. If such a single muscle stimulation pattern is used in reality, the control problem is reduced to starting the movement at the right instant of time.
Although this constitutes a problem of its own, it should be
manageable because of the direct visual perception of time
to contact (Lee 1976).
We gratefully acknowledge Prof. Dr. K. van der Werff, Delft University
of Technology, for giving us full access to the source code of SPACAR.
Address for reprint requests: A. J. van Soest, Faculty of Human Movement Sciences, van der Boechorststraat 9, NL 108 1 BT Amsterdam, The
Netherlands.
Received 22 January 1993; accepted in final form 29 November 1993.
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