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Control of maximal and submaximal vertical jumps
van Zandwijk, J.P.; Bobbert, M.F.; Munneke, M.
published in
Medicine and Science in Sports and Exercise
2000
DOI (link to publisher)
10.1097/00005768-200002000-00033
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citation for published version (APA)
van Zandwijk, J. P., Bobbert, M. F., & Munneke, M. (2000). Control of maximal and submaximal vertical jumps.
Medicine and Science in Sports and Exercise, 32(2), 477-485. https://doi.org/10.1097/00005768-200002000-
00033
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Control of maximal and submaximal
vertical jumps
JAN PETER VAN ZANDWIJK, MAARTEN F. BOBBERT, MARTEN MUNNEKE, and PIETER PAS
Institute for Fundamental and Clinical Human Movement Sciences, Vrije Universiteit, Amsterdam, THE NETHERLANDS
ABSTRACT
VAN ZANDWIJK, J. P., M. F. BOBBERT, M. MUNNEKE, and P. PAS. Control of maximal and submaximal vertical jumps. Med.
Sci. Sports Exerc., Vol. 32, No. 2, pp. 477–485, 2000. Purpose: It was investigated to what extent control signals used by human
subjects to perform submaximal vertical jumps are related to control signals used to perform maximal vertical jumps. Methods: Eight
subjects performed both maximal and submaximal height jumps from a static squatting position. Kinematic and kinetic data were
recorded as well as electromyographic (EMG) signals from eight leg muscles. Principal component analysis was used analyze the shape
of smoothed rectified EMG (SREMG) histories. Jumps were also simulated with a forward dynamic model of the musculoskeletal
system, comprising four segments and six muscles. First, a maximal height jump was simulated by finding the optimal stimulation
pattern, i.e., the pattern resulting in a maximum height of the mass center of the body. Subsequently, submaximal jumps were simulated
by adapting the optimal stimulation pattern using strategies derived from the experimental SREMG histories. Results: SREMG
histories of maximal and submaximal jumps revealed only minor differences in relative timing of the muscles between maximal and
submaximal jumps, but SREMG amplitude was reduced in the biarticular muscles. The shape of the SREMG recordings was not much
different between the two conditions, even for the biarticular muscles. The simulated submaximal jump resembled to some extent the
submaximal jumps found in the experiment, suggesting that differences in control signals as inferred from the experimental data could
indeed be sufficient to get the observed behavior. Conclusions: The results fit in with theories on the existence of generalized motor
programs within the central nervous system, the output of which is determined by the setting of parameters such as amplitude and
relative timing of control signals. Key Words: HUMAN JUMPING, ELECTROMYOGRAPHIC ANALYSIS, MATHEMATICAL
MODELING, OPTIMIZATION
H
uman subjects are able to execute most motor tasks
at different levels of performance. When a given
task is executed maximally, the subject attempts to
achieve the highest performance possible. On the other
hand, when submaximal performance is asked for, the sub-
ject attempts to attain a certain level of performance, which
may be prescribed by the experimenter. It is the task of the
central nervous system (CNS) to generate in each case an
appropriate set of control signals to all muscles involved in
execution of the task. In case of performing a task maxi-
mally, this might be relatively easy from a control point of
view, because there exists a unique set of control signals
yielding maximal performance. These optimal control sig-
nals can be the result of some learning process in which
controls are adapted over time to yield finally those giving
maximal performance. Providing control signals for sub-
maximal performance of a task is, however, more difficult
for a number of reasons. In the first place, there exist, in
principle, different sets of control signals which all yield the
same submaximal performance. Besides this, there are many
levels of submaximal performance possible, each of which
requires an appropriate set of control signals.
On the basis of these considerations, it remains an in-
triguing puzzle how the CNS generates control signals for
different levels of performance of a motor task. It seems
unlikely that the CNS explicitly calculates suitable control
signals for each level of performance using some internal
representation, because most motor tasks can be initiated
almost instantaneously. Also, it does not appear to be a
feasible option that control signals for each level of sub-
maximal achievement of a motor task are stored somewhere
in the CNS in the form of a motor program. In that case,
retrieval of the appropriate motor program for each level of
performance would be a problem. Besides this, it would be
difficult to explain successful performance at new levels of
performance. An elegant alternative which circumvents the
storage and novelty problem is based on the concept of
generalized motor programs (9). A generalized motor pro-
gram is a template motor program for a particular class of
movements, the output of which is determined by the setting
of certain parameters. Once a certain rule is provided to
adjust parameters within the generalized motor program,
this program can be used to provide control signals for both
maximal performance of a task and all levels of submaximal
performance.
This paper addresses the issue how the CNS generates
control signals in case of multi-joint vertical squat jumping
0195-9131/00/3202-0477/0
MEDICINE & SCIENCE IN SPORTS & EXERCISE
®
Copyright © 2000 by the American College of Sports Medicine
Submitted for publication December 1997.
Accepted for publication December 1998.
477
to different heights. Vertical jumping belongs to the class of
explosive movements. These movements are characterized
by a short execution time and are aimed at giving a high
velocity to a part of the body. Because of the short execution
time, afferent feedback can only play a limited role in the
control of such movements. This means that control signals
must to a large extent be preprogrammed and that therefore
controlling such movements relies heavily on storage ca-
pacity of the CNS. Furthermore, vertical squat jumping is
attractive for studying movement control because perfor-
mance can be unambiguously defined in terms of jump
height. Since the focus will be on general organizing prin-
ciples in the control of explosive movements, it will first be
investigated whether different subjects consistently perform
submaximal vertical squat jumps in a similar way. Sec-
ondly, differences in control signals between maximal and
submaximal squat jumping will be analyzed to see whether
control signals for these two levels of performance are
related. For this purpose, one requires a measure for control
signals to each muscle involved in the execution of the
movement.
Although measures for neural control signals cannot be
obtained directly, one can record electromyographical sig-
nals (EMG signals) from active muscles in human subjects.
Despite the fact that these EMG signals are electrical out-
puts of muscle, they are closely related to neural control
signals to the muscles (e.g., 6,8,15). Therefore, in order to
investigate the control strategy employed during the execu-
tion of maximal and submaximal squat jumping, EMG
signals recorded during maximal and submaximal squat
jumping are compared. Differences observed in EMG sig-
nals between the two conditions could be the result of
parameter adjustment in the generalized motor program
used in the execution of squat jumping. In an simulation
study, van Soest and Bobbert (10) proposed a control strat-
egy for generating control signals in case of submaximal
squat jumping, which results in scaling of net joint moments
and hence identical kinematics at different speeds of move-
ment. Such a control strategy would provide the advantage
that performance remains predictable.
Finally, it will be examined whether the differences in
EMG signals found between maximal and submaximal
jumping are sufficient for obtaining submaximal perfor-
mance by means of numerical simulation of the push-off
phase in vertical squat jumping. To this end, control signals
pertaining to a maximal height squat jump in a model of the
human musculo-skeletal system are adapted according to
the differences in EMG signals found between maximal and
submaximal jumping, and it will be examined to what extent
the resulting jumps in the model resemble submaximal squat
jumps found in human subjects.
METHODS
Subjects
Eight male volunteers (age 26 ⫾ 3 yr, height 1.91 ⫾
0.05 m, body mass 83 ⫾ 7 kg) participated in this study.
Informed consent was obtained from each subject according
to the policy statement of the American College of Sports
Medicine.
Protocol
Each subject performed maximal and submaximal jumps
from the same static squatting position. To help the subject
reproduce the same initial position each time a device was
used which consists of two boards fixed to a pole in a hinge.
The angle of the boards with the pole as well as the height
of hinge can be varied independently. First, the subject
assumed a freely chosen initial position. In this position, the
angles of both boards and the height of the hinge were set to
match hip and knee segment angles and height of the hip
joint as closely as possible. It is easily shown that once these
three parameters are fixed the initial position is determined
unambiguously.
Before all subsequent jumps, subjects adjusted their ini-
tial position to the device to match the initial position of the
first jump as accurately as possible. After this adjustment,
the device was pulled back by the experimenter and the
subject performed the jump. The subject was instructed keep
his arms crossed behind his back during execution of the
jumps, to jump without making preparatory countermove-
ment and to initiate the jump as soon as possible after a beep
signal. All subjects performed both maximal height and
submaximal height jumps. In the case of maximal height
jumping, the subjects were instructed to jump as high as
possible. In the case of submaximal jumping, a target height
was indicated by means of a small light source that was
placed at some distance behind a narrow slit. The light
source could only be seen when looked at horizontally
through the slit. Subjects were instructed to jump to such a
height, that they could just see the lightsource. This proce-
dure ensured that the subjects attained about the same jump
height each time they performed a submaximal jump. Jump
height is defined as the height reached by the centre of mass
(CM) of the body at the apex of the jump relative to the
height of the CM of the body in upright standing. Figure 1
schematically shows the setup used.
After some practice jumps, each subject performed three
maximal height jumps from which averaged maximal jump
height was calculated. Subsequently, the light source was
placed at a height corresponding to approximately 75% of
maximal jump height. By choosing such a high percentage,
it was hoped that control of the movement remained open
loop in case of submaximal jumping, which might not be the
case if a smaller percentage of maximal jump height was
selected. Next, each subject performed six maximal and
eight submaximal jumps in random order.
Kinematics and Kinetics
In this study reflecting markers were placed on fifth
metatarsophalangeal joint, calcaneus, lateral malleolus,
knee joint (on the lateral collateral ligament at the height of
the joint cleft), greater trochanter, and neck (at the height
of the fifth cervical vertebra). These markers defined the
478
Official Journal of the American College of Sports Medicine http://www.msse.org
position of the four body segments: feet, lower legs, upper
legs, and head-arms-trunk (HAT). During jumping kine-
matic data were obtained using high speed video (VICON,
Oxford Metrics Ltd.) at a sample rate of 100 Hz. Simulta-
neously, vertical and fore-aft components of the ground
reaction force and its point of application were measured
using a force platform (Kistler 9281B, Kistler Instruments
Corp., Amherst, NY) and sampled at 200 Hz.
Electromyography
Electromyographic signals (EMG signals) of eight mus-
cles of one leg were recorded during the execution of the
jumps using pairs of surface electrodes (Meditrace ECE
1801) after standard skin preparation techniques (2). The
muscles selected were lateral and medial head of m. gas-
trocnemius, m. soleus, m. semitendinosus, long head of m.
biceps femoris, m. vastus lateralis, m. rectus femoris, and m.
gluteus maximus. The electrical signals of the muscles were
amplified (Disa 15 C01, Disa Electronics, Skovlunde Den-
mark) and 7-Hz high-pass filtered to eliminate movement
artifacts. Subsequently the electrical signals were rectified,
22-Hz low-pass filtered and sampled at 200 Hz, yielding
smoothed rectified EMG signals (SREMG signals).
Treatment of Data
For each subject, the three highest maximal jumps and the
three lowest submaximal jumps were selected for further
analysis. Kinematic and kinetic variables of different jumps
were synchronized at the instant the subject left the ground
(subsequently referred to as toe-off) and truncated to contain
only the last 750 ms of the push-off phase before averaging.
The SREMG recordings were synchronized the same way
and additionally for each trial baseline activity (i.e., activity
of the muscles before the jump was executed) was sub-
tracted before averaging.
Electromyographic Data Analysis
Differences in control signals to the muscles between
maximal and submaximal jumps may consist of a combi-
nation of (i) a change in amplitude of control signals to the
muscles, (ii) a change in shape of control signals to the
muscles, and (iii) a change in relative timing of control
signals to the muscles. So the SREMG recordings of the
averaged maximal and submaximal jumps were searched for
all of these possibilities, using the following methods:
Amplitude of the control signals. Differences in am-
plitude of the control signals to the muscles were quantified
by computing the ratio of the time integrals of the SREMG
histories of the averaged submaximal jump to those of the
averaged maximal jump. So if a muscle is less active in case
of submaximal jumping, this will lead to a ratio which is
smaller than one. Subsequently, for each muscle these ratios
were averaged across subjects and it was tested whether the
averaged ratio differed significantly from 1.0 using a Stu-
dent t-test for paired comparisons at a level of significance
of 5%.
Shape of the control signals. To quantify the differ-
ence in shape of the control signals to the muscles in
maximal and submaximal jumping principal component
analysis (PCA) was performed on averaged maximal and
submaximal SREMG histories for each muscle (see also:
3,4). This statistical technique computes from a set of data
waveforms {s
i
} a set of orthonormal principal component
waveforms {pc
j
} and a set of weighting coefficients {c
ij
},
such that
s
i
⫽
冘
j
c
ij
pc
j
@i (1)
By definition, the first principal component is the best mean
square representation of all data waveforms in the set {s
i
},
the second principal component is the best mean square
representation to the data waveforms {s
i
} after the first
component has been subtracted, and so on. The fraction f
1
of
the variance of the set {s
i
} explained by the first principal
component equals
f
1
⫽
冘
i
c
i1
2
冘
ij
c
ij
2
(1)
If there is a large difference in shape of control signals
between maximal and submaximal jumping, this is re-
flected in a small fraction of variance explained by the
first principal component. Before PCA, mean values were
subtracted from the SREMG histories, and since in this
part of the analysis we are only interested in differences
in shape of control signals to the muscles and not in
differences in amplitude, maximal and submaximal
SREMG histories were normalized to unit variance. After
PCA for each muscle the fractions found were averaged
across subjects.
Figure 1—Schematic view of the setup used in the experimental study.
a, Device used to help subject reproduce the same starting position
each time they produced a vertical squat jump. b, Apparatus contain-
ing a light source to indicate target height in case of submaximal
jumping.
MAXIMAL AND SUBMAXIMAL JUMP CONTROL Medicine & Science in Sports & Exercise
姞
479
Relative timing of control signals. To detect differ-
ences in relative timing of control signals to the muscles, the
onset of activity for each muscle was determined for both
maximal and submaximal jumping. The onset of activity
was taken as the instant of first sustained rise of the SREMG
above the baseline. The shift in onset time for each muscle
was averaged across subjects and it was tested whether the
averaged shift differed significantly from zero using a Stu-
dent t-test for paired comparisons at a level of significance
of 5%.
Computer Simulations Using a Model of the
Human Musculoskeletal System
Computer simulations of the push-off phase of a vertical
squat jump were performed using a model of the human
musculoskeletal system which has already been described in
detail elsewhere (e.g., 1,11). In short, the model consists of
four rigid segments, representing feet, lower legs, upper
legs, and upper body, connected in frictionless hinge joints.
Six important muscle groups for extension of the lower
extremities (m. gastrocnemius, m. soleus, hamstrings, mm.
vasti, m. rectus femoris, and m. gluteus maximus) are in-
corporated into the model by means of Hill-type muscle
models. Each muscle model consists of two sets of equa-
tions, one describing the contractile behavior of muscle, the
other describing its excitation by the central nervous system.
The former will be called the contraction dynamics of the
muscle model, the latter its excitation dynamics. For the
human calf muscles, parameter values for both the excita-
tion and contraction dynamics are available which have
been determined on the basis of experimental data obtained
from these muscles (14). Numerical techniques used for this
purpose have been evaluated first for rat isolated skeletal
muscle before being used on data from human muscle (see
e.g., 12,13). Because presently data pertaining to both ex-
citation and contraction dynamics are not available for other
muscle groups than m. triceps surae, it was decided to use
these parameter values for all six muscle groups incorpo-
rated in the model. Input to the model is stimulation to each
of the six muscles, i.e., a number between 0 and 1 being a
one-dimensional representation of recruitment and firing
rate of the a motoneurons (5). Among the output of the
model is movement of the body segments.
Besides excitation and contraction dynamics, the dynam-
ics of neural control signals can be a functional factor in the
control of movement. These dynamics will be referred to as
stimulation dynamics. For isometric contractions of the calf
muscles, it was shown in (15) that stimulation dynamics was
a functional factor influencing the rate of muscle moment
development. The effect of stimulation dynamics was in-
corporated into the model by letting control signals to all
muscles rise at a finite rate to their final values. This rate
was chosen to be the same for all muscles and corresponded
to the rate of change of the averaged SREMG signals during
maximal jumping, averaged over all muscles. In the simu-
lations, the stimulation to each muscle was allowed to
change only once from its initial value to its maximal value
of 1 and was forced to remain maximal during the rest of the
simulation. This reduced the control problem of vertical
jumping to finding that combination of six muscle stimula-
tion onset times which yielded the highest performance in
terms of jump height. The numerical experiments consisted
in the first place of finding by means of numerical optimi-
zation that combination of onset times of the stimulation to
the six muscles which yields the highest jump. Secondly, the
stimulation to the muscles in the model was adapted ac-
cording to the differences in SREMG signals between sub-
maximal and maximal jumps as observed in the experi-
ments. Finally, performance of the model using these new
control signals was evaluated.
RESULTS
Experimental Data
In this section, the focus will be on the data of the vertical
ground reaction force and SREMG recordings, since the
former directly relates to the movement of the CM of the
body and thus to performance and the latter is a measure for
control signals to the muscles. Table 1 shows jumping
parameters of the maximal and submaximal jump averaged
across subjects. The difference between maximal and sub-
maximal jump height amounted to 8 cm on average. Figure
2 shows for one subject stick diagrams of the initial position
and the position at toe-off. In Figure 2 as well as in the
remainder of this paper, solid curves pertain to averaged
maximal jumps and dashed curves to averaged submaximal
jumps. From Table 1 and Figure 2, it is apparent that
subjects were able to reproduce the same initial position
fairly well using the device shown in Figure 1. Also, it is
interesting to observe that in case of submaximal jumping
hip and knee joints are extended less at toe-off. The angular
displacement of hip and knee joint (i.e., the difference
between joint angle at toe-off and initial joint angle) was
found to be significantly less (P ⬍ 0.05) in submaximal
jumping than in maximal jumping. For the ankle joint no
significant difference in angular displacement was found
between maximal and submaximal jumping. Figure 3 shows
for the same subject the vertical ground reaction force for
TABLE 1. Jumping parameters of maximal and submaximal jumps.
Parameter
Maximal
Jump
Submaximal
Jump
jump height [m] 0.39 ⫾ 0.05 0.31 ⫾ 0.04
V
cm
toe-off
[m䡠s
⫺1
]
2.4 ⫾ 0.2 2.1 ⫾ 0.3
F
peak
[
N
]
2100 ⫾ 400 2100 ⫾ 400
h
initial
1.4 ⫾ 0.2 1.4 ⫾ 0.2
h
toe-off
2.9 ⫾ 0.1 2.8 ⫾ 0.1
k
initial
1.7 ⫾ 0.2 1.7 ⫾ 0.3
k
toe-off
3.0 ⫾ 0.1 2.9 ⫾ 0.2
a
initial
1.5 ⫾ 0.1 1.5 ⫾ 0.1
a
toe-off
2.6 ⫾ 0.1 2.4 ⫾ 0.3
V
cm
toe-off
, vertical velocity of the CM that the instant of toe-off; F
peak
, maximal value
attained by the vertical ground reaction force during the push-off phase;
h
initial
, initial
hip angle;
h
toe-off
, hip angle at the instant of toe-off;
k
initial
, initial knee angle;
k
toe-off
,
knee angle at the instant of toe-off;
a
initial
, initial ankle angle;
a
toe-off
, ankle angle at the
instant of toe-off.
For each joint full extension corresponds to
radians. The parameter values
shown (mean ⫾ SD) are averages across subjects (
N
⫽ 8).
480
Official Journal of the American College of Sports Medicine http://www.msse.org
