J Neurophysiol 89: 1784 –1796, 2003.
First published December 27, 2002; 10.1152/jn.00674.2002.
Utilization and Compensation of Interaction Torques During Ball-Throwing
Movements
Masaya Hirashima, Kazutoshi Kudo, and Tatsuyuki Ohtsuki
Department of Life Sciences (Sports Sciences), Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
Submitted 13 August 2002; accepted in final form 14 December 2002
INTRODUCTION
Most human and animal movements include the rotations of
two or more joints. In multijoint limb movements, torque at
one joint occurs not only from muscles acting at that joint but
also from interactions due to the rotations of other joints
(Hollerbach and Flash 1982). The net torque around one joint,
which is proportional to the angular acceleration at the joint, is
represented as the sum of the muscle torque, gravity torque,
and interaction torque. Therefore the CNS is required to generate appropriate motor commands taking all these torques into
consideration to execute the desired multijoint movement. The
interaction torque is one of the self-generated load that is
generated by the movement itself and is absent before the
motion. Recent studies have demonstrated the ability of humans to predict and compensate for the self-generated loads
such as the Coriolis force (Cohn et al. 2000; DiZio and Lackner
1995, 2000, 2001; Lackner and DiZio 1994), the tangential
load force in lifting or transporting the object (Flanagan and
Wing 1993, 1995, 1997; Johansson and Westling 1984, 1988;
Johansson et al. 1992a,b), the back force from the ball in
throwing (Hore et al. 1999, 2001), and the interaction torque in
multijoint movements (Almeida et al. 2000; Beer et al. 2000;
Grible and Ostry 1999; Koshland et al. 2000; Sainburg 2002;
Address for reprint requests: M. Hirashima, c/o Dr. Tatsuyuki Ohtsuki,
Dept. of Life Sciences (Sports Sciences), Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan
153-8902 (E-mail: [email protected]).
1784
Sainburg and Kalakanis 2000; Sainburg et al. 1993, 1995,
1999; Topka et al. 1998). Studies on patients with nervoussystem injury clearly indicated that the compensation for the
interaction torque is essential for the execution of the accurate
multijoint movements. In reaching movement, for example,
cerebellar patients could not deal with the interaction torque
appropriately, and, as a consequence, an abnormally curved
hand path was produced (Bastian et al. 1996).
Ball-throwing is one of the most skilled multijoint movements requiring excellent coordination between joints. Humans
can throw a ball with wide range of ball speed keeping hitting
accuracy. To accomplish this, the CNS must produce the
appropriate motor command predicting the back forces from
the ball and the interaction torque at each joint. Hore et al.
(2001) examined skilled throwers’ ability to predict the back
forces by instructing them to throw balls of different weights
and diameters with different speeds. The skilled throwers could
adjust the finger grip force in proportion to the back forces and
could keep the amplitude of finger extension relatively constant
from throw to throw. In addition, Timmann et al. (2001)
showed that cerebellar patients and unskilled throwers could
not control the finger grip force or finger extension precisely.
These findings show that the CNS of the skilled ball-thrower
could predict and compensate for the back force that is caused
by the hand acceleration.
This hand acceleration is generated by the proximal-to-distal
sequence of joint rotations (Putnam 1993) that is produced by
the proximal-to-distal sequential muscle activities (Hirashima
et al. 2002). However, it is unclear how the CNS deals with the
interaction torque at each joint during throwing. The purpose
of this study is to examine the general feature of the interaction
torque that is determined by the mechanics or the trajectories of
the arm in throwing and to determine how the CNS makes use
of or compensates for the interaction torque at each joint to
throw a ball with the desired speed and accuracy. We examine
the intersegmental dynamics in the upper extremity by instructing subjects to throw a ball with different speeds (slow, medium, and fast) in random order. Because the intersegmental
dynamics is greatly influenced by the joint angular velocity and
angular acceleration (Hollerbach and Flash 1982), to throw a
ball with desired speed, it is essential for the CNS to precisely
predict and compensate for the interaction torque at each joint
from throw to throw. In addition, because the number of joints
mobilized in throwing affects the intersegmental dynamics, we
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Hirashima, Masaya, Kazutoshi Kudo, and Tatsuyuki Ohtsuki.
Utilization and compensation of interaction torques during ball-throwing movements. J Neurophysiol 89: 1784 –1796, 2003. First published
December 27, 2002; 10.1152/jn.00674.2002. The manner in which the
CNS deals with interaction torques at each joint in ball throwing was
investigated by instructing subjects to throw a ball at three different
speeds, using two (elbow and wrist) or three joints (shoulder, elbow,
and wrist). The results indicated that the role of the muscle torque at
the most proximal joint was to accelerate the most proximal joint and
to produce the effect of interjoint interaction on the distal joints. In the
three-joint throwing, shoulder muscle torque produced the assistive
interaction torque for the elbow, which was effectively utilized to
generate large elbow angular velocity when throwing fast. However,
at the wrist, the muscle torque always counteracted the interaction
torque. By this kinetic mechanism, the wrist angular velocity at the
ball-release time was kept relatively constant irrespective of ball
speed, which would lead to an accurate ball release. Thus it was
concluded that humans can adjust the speed and accuracy of ballthrowing by utilizing interaction torque or compensating for it.
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
compare the two-joint throwing (elbow and wrist) with threejoint throwing (shoulder, elbow, and wrist). Thus we are able
to investigate general feature of the intersegmental dynamics of
throwing and its changes caused by the differences of the ball
speed and the number of mobilized joints. We also discuss the
strategy the CNS would use to adjust the ball speed and to
release the ball accurately.
METHODS
Subjects
Apparatus and protocols
This study examined two types of throwing: “two-joint throwing,”
which includes elbow and wrist joint rotations (8 subjects), and
“three-joint throwing,” which includes shoulder, elbow, and wrist
joint rotations (7 subjects). Reflective markers were attached to the
shoulder, elbow, wrist, and metacarpophalangeal (MP) joints of the
right side of the right arm. During two-joint throwing, subjects were
instructed to throw without translation of the elbow. During threejoint throwing, subjects were instructed to throw, as much as possible,
without translation of the shoulder. Because actually there were some
translations, we took the translation into consideration in calculating
the inverse dynamics (see APPENDIX).
We instructed subjects to throw a straight ball with all markers in
one vertical plane throughout the movement without pronation or
supination of the forearm and not to throw a curve ball. Some subjects
started with two-joint throwing; the others started with three-joint
throwing. Throws were made under three different speed conditions:
“slow-accurate,” “medium-accurate,” and “fast-accurate.” Each of
three conditions was randomly presented once to construct one bout.
Ten bouts were repeated. Thus in each condition, subjects made 10
throws. Each subject made a total of 60 throws (30 for 2-joint and 30
for 3-joint). Additional care was taken so that the same condition was
not repeated consecutively to force subjects to predict and compensate
for the interaction torque from throw to throw stronger than in the case
of repeating same speed. After a warm-up, subjects threw a baseball
using the right hand toward a target on a wall (Fig. 1). The subjects
were instructed to put the arm prior to the throw in the initial position
in which the passive torque from the connective tissues of the arm
balanced with gravity torque (Fig. 1). The horizontal distance between
the wall and the elbow joint in two-joint throwing was 2.7 m (Fig. 1A).
The horizontal distance between the wall and the shoulder joint in
three-joint throwing was 3.2 m (Fig. 1B). The height of the center of
the target was 0.2 m above the height of the eye. The target board was
a 1 ⫻ 1-m square. When the ball did not hit the target board, the trial
was repeated.
Kinematic analysis
Motions of the markers were recorded at 200 Hz using two highspeed video cameras (HAS-200R, Ditect). By using the direct linear
transformation (DLT) method, time-position data in three dimensions
were obtained. We considered the projections of markers to the plane
that is determined by the three points (shoulder, elbow, and wrist)
when the elbow-joint angle was right angle. Angular displacements at
each joint were obtained in this plane. Joint angles are shown in Fig.
1. We did not adopt “segment angles” (Hoy et al. 1985) but instead
adopted the conventional “joint angles” of Hollerbach and Flash
(1982). Cooper et al. (2000) and Sainburg et al. (1995) recommended
joint angles because the nervous system has proprioceptive information about the joint motion but lacks a direct measure of segment
angles. Therefore the equations of motion in terms of joint angles
would more accurately represent the control problem from the viewpoint of the brain.
The direction of the arrow in the stick pictures in Fig. 1 represents
the positive direction of the angular displacement. For example, the
positive direction of the ␪1 represents the elbow flexion during twojoint throwing. Note that the throwing motion consists of the elbow
extension (negative direction) and wrist flexion (negative direction)
for two-joint throwing (Fig. 2B), and the shoulder extension (negative
direction), elbow extension (negative direction) and wrist flexion
(negative direction) for three-joint throwing (Fig. 4B).
FIG. 1. Experimental setup and joint angles in
2-joint throwing (A) and 3-joint throwing (B). The
positive direction of each angular displacement is the
counterclockwise direction. Note that the throwing
motion consists of the elbow extension (negative
direction) and wrist flexion (negative direction) for
2-joint throwing, and the shoulder extension (negative direction), elbow extension (negative direction),
and wrist flexion (negative direction) for 3-joint
throwing.
J Neurophysiol • VOL
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Eight healthy right-handed males (mean age: 23.8 yr) volunteered
as subjects in this study. Subjects were clearly informed of the
procedures of the experiment, according to the Declaration of Helsinki
and gave a written informed consent before the experiment. This
experimental procedure was approved by the Ethical Committee of
the Graduate School of Arts and Sciences of the University of Tokyo.
The subjects had no musculoskeletal disorders and were in excellent
physical condition. There were two baseball players among the subjects (subjects A and B). However, they were not treated separately
because their feature of the intersegmental dynamics was similar to
that of the other subjects, and it was not the purpose of this study to
identify the difference between skilled and unskilled throwing.
1785
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M. HIRASHIMA, K. KUDO, AND T. OHTSUKI
Slow
A
Medium
Fast
Ball
20 cm
MP
Wrist
Ball
Release
Elbow
B
Angle [deg]
240
220
200
180
160
140
Elbow
Extension
240
220
200
180
160
140
240
220
200
180
160
140
Wrist
Flexion
40
20
0
-20
-40
-60
40
20
0
-20
-40
-60
Anglular Velocity
[deg. s-1]
-200
C
20 cm
-100
0 [ms]
-200
0
-500
-100
0 [ms]
0
0
-500
-500
-1000
-1000
Elbow
-200
-100
0 [ms]
-200
-100
0 [ms]
-200
-100
0 [ms]
Extension
Wrist
-1000
Flexion
-200
-100
0 [ms]
-200
-100
0 [ms]
Wrist
Torque [N m]
D
.
2
Elbow
Torque [N m]
2
0
0
-2
-2
Flexion 10
10
0
0
-10
-10
0
Flexion
-2
10
.
2
Extension
Net
Gravity
Muscle
Interaction
0
Extension
-10
-20
-20
-200
-100
-20
0 [ms]
-200
-100
0 [ms]
FIG.
2. Data for representative single trials in 2-joint throwing for slow (left), medium (middle), and fast condition (right)
from subject A are shown. A: stick pictures at ⫺100, ⫺75, ⫺50, ⫺25, and 0 ms are drawn. The target is in the right direction.
B: joint angles at the wrist (dashed line) and elbow (solid line). C: joint angular velocity at the wrist (dashed line) and elbow
(solid line). D: the net torque (solid line), gravity torque (dashed line), muscle torque (gray line), and interaction torque
(dotted line) at each joint. The vertical lines at 0 ms represent the ball-release time. Four vertical dotted lines represent ⫺100,
⫺75, ⫺50, and ⫺25 ms.
Angular displacement data were smoothed with the use of the
bidirectional fourth-order Butterworth low-pass filter (cut-off frequency ⫽ 13 Hz) and were numerically differentiated to obtain
angular velocity and angular acceleration. The onset of the movement
was detected as the time when the angular velocity at the most
proximal joint exceeded 10% of its maximum.
J Neurophysiol • VOL
Kinetic analysis
We employed inverse dynamics and calculated the time series of
the net torque (NET), the gravity torque (GRA), the interaction torque
(INT), and the muscle torque (MUS) about the shoulder, elbow, and
wrist joints. Complete mathematical equations are presented in the
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40
20
0
-20
-40
-60
Ball
Release
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
APPENDIX.
Index of coordination between the interaction torque and the
muscle torque (IOCIM)
There are two representative coordination relationships between the
interaction torque and the muscle torque. One is the “counteractive”
relationship, which is always seen at the wrist joint during reaching
movements (Dounskaia et al. 1998; Galloway and Koshland 2002;
Ghez et al. 1996; Koshland et al. 2000; Virji-Babul and Cooke 1995).
In counteractive relationship, the muscle torque plays a counteractive
role against the interaction torque and keeps the net torque relatively
small throughout the movement. The counteractive relationship is also
seen during the single-joint instructed movements (Almeida et al.
1995; Bastian et al. 2000; Galloway and Koshland 2002; Gribble and
Ostry 1999) in which subjects were instructed to move only one joint
(focal joint) without moving other joints (nonfocal joints). Subjects
kept the nonfocal joints motionless during the motion of the focal joint
by the muscle torque appropriately counteracting the interaction
torque at the nonfocal joints.
The other is the “assistive” relationship, which is sometimes seen at
the elbow joint during reaching movements (Galloway and Koshland
2002). In the assistive relationship, the muscle torque and the interaction torque work together for the same direction to generate larger
net torque. This assistive relationship is advantageous for generating
large angular velocity at the joint.
To quantify intersegmental dynamics, we made the index of coordination between the interaction torque and the muscle torque
(IOCIM) so that this index can distinguish the “counteractive” or
“assistive” relationship. IOCIM for one trial was calculated as follows
IOCIM ⫽
T2
I共t兲dt
兰 T1
T2
兰T1 M共t兲dt
where T1 is the time of movement onset at the most proximal joint and
T2 is the ball-release time, I(t) and M(t) was calculated as follows
I共t兲 ⫽
再
⫺兩INT共t兲兩
⫹兩INT共t兲兩
共INT共t兲 䡠 MUS共t兲 ⬍ 0兲
共INT共t兲 䡠 MUS共t兲 ⱖ 0兲
M共t兲 ⫽ ⫹兩MUS共t兲兩.
The positive sign of IOCIM indicates the “assistive” relationship,
and the negative sign of IOCIM indicates the “counteractive” relationship from T1 to T2 as a whole. The magnitude of the IOCIM
J Neurophysiol • VOL
indicated the relative magnitude of the interaction torque to the
muscle torque. For example, IOCIM value of ⫹1.0 occurs when the
interaction torque and the muscle torque have the same direction and
the same amount of torque impulse; ⫺1.0 occurs when they have the
opposite direction and the same amount of torque impulse; ⫺0.5
occurs when they have the opposite direction and the interaction
torque impulse is half of the muscle torque impulse.
Statistics
Repeated-measures ANOVAs were performed to assess the effect
of the speed on the MUS impulse, INT impulse, NET impulse, and
angular velocity at the ball-release time (P ⬍ 0.05). Tukey’s post hoc
multiple comparison tests determined the increase or decrease of these
variables between speeds (P ⬍ 0.05).
RESULTS
The results consist of two main sections. The first section
shows the general features of the kinematics and kinetics in
throwing movements, especially focusing on the relationship
between the interaction torque and muscle torque. These features are necessary to understand the mechanisms of the strategy to adjust the ball speed that is clarified in the second
section. Each section consists of two subsections, i.e., two- and
three-joint throwing.
General features of the kinematics and kinetics in throwing
movements
TWO-JOINT THROWING. Figure 2A shows stick pictures of one
trial for subject A, when throwing balls at three different speeds
(slow, medium, and fast) with two joints. Figure 2, B and C,
shows the kinematics of the same trials of Fig. 2A. Figure 2D
shows the kinetics of them. The ball-release time was set to 0
ms. First the elbow started to extend (slow, ⫺235 ms; medium,
⫺195 ms; Fast, ⫺160 ms; see Fig. 2C). Next the wrist started
to flex (about ⫺80 ms), and the ball was released (0 ms).
Elbow. For all speeds, before the movements, the elbow
extension muscle torque was kept at a magnitude that opposed
the flexion gravity torque, and it held the elbow in the starting
position (Fig. 2D). Next, from about ⫺200 to ⫺100 ms, the
subject increased the elbow extension muscle torque to produce the elbow extension net torque that is proportional to the
angular acceleration of the elbow. This elbow extension net
torque initiated the two-joint throwing (Fig. 2C). In this initial
phase, the elbow interaction torque had little effect at the
elbow. Next, from about ⫺100 to 0 ms, the elbow flexion
interaction torque occurred and counteracted the elbow muscle
torque almost throughout this phase.
To clearly determine whether there was the counteractive or
assistive relationship, the interaction torque was plotted against
the muscle torque from the movement onset to the ball-release
time for the same trial of Fig. 2 (see Fig. 3A). In the elbow joint
in Fig. 3A, most of the plots exist in the second quadrant. This
means that the interaction torque and the muscle torque always
showed different signs, i.e., the “counteractive” relationship.
The IOCIM (see METHODS) is also shown in Fig. 3A. All
IOCIMs for the elbow showed negative sign (i.e., the counteractive relationship). Figure 3B shows the interaction torques of
all trials for all subjects plotted against their muscle torques for
two-joint throwing. Averages of IOCIM across all subjects are
also shown. They all show negative sign.
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Anthropometric data are based on the data of Japanese
athletes by Ae et al. (1992), which appears on the paper by Yokoi et
al. (1998). With respect to these four torques, clear definitions and
in-depth explanations have been given in several studies (Bastian et al.
1996, 2000; Cooper et al. 2000; Hollerbach and Flash 1982).
Bastian et al. (1996) calculated the NET as the product of the
moment of inertia of the involved segments (including the segment
under consideration and all segments distal to it) and the angular
acceleration around the joint under consideration. We adopted this
definition. As the NET is defined as the sum of other components, it
is described as follows: NET⫽ GRA ⫹ INT ⫹ MUS. The GRA is the
term with the gravitational acceleration.
The INT at each joint is the sum of the terms with the angular
accelerations of the other joint (inertial torque), the terms with the
product of the angular velocities of the same joint (centripetal torque),
the terms with the product of the angular velocities of the different
joints (Coriolis torque), and the terms with the linear acceleration of
the most proximal joint (see Hollerbach and Flash 1982).
As the MUS is calculated as residual terms as follows: MUS ⫽
NET ⫺ GRA ⫺ INT, the MUS is sometimes called “residual torque.”
It is noted that the MUS includes not only the mechanical contribution
of muscle contraction acting at the joint but also the passive contributions by muscles, tendons, ligaments, articular capsules, and other
connective tissues.
1787
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M. HIRASHIMA, K. KUDO, AND T. OHTSUKI
A
Slow
Medium
INT [N m]
Wrist
4
.
Fast
4
-0.47
2
4
-0.58
2
-0.55
2
Ext.
0
0
0
-2
-2
-2
-4
-4
-4
-4
-4
-4
4
-2
0
2
4
Flex.
Flex.
-2
0
2
4
.
-2
0
2
MUS [N m]
Ext.
INT [N m]
Elbow
20
.
20
-0.09
20
-0.06
10
10
10
0
0
0
-10
-10
-10
-20
-20
0
20
MUS [N m]
Flex.
Flex.
Ext.
-20
-20
0
20
FIG. 3. A: interaction torque plotted against
the muscle torque from movement onset to ball
release of the same trial as Fig. 2. The index of
coordination between INT and MUS (IOCIMs)
are also shown. B: interaction torque plotted
against the muscle torque from movement onset
to ball release of all trials for all subjects in
2-joint throwing. The IOCIMs averaged for all
subjects are also shown.
Ext.
-20
0
20
.
Slow
B
Medium
INT [N m]
Wrist
4
.
Fast
4
-0.38
2
4
-0.44
2
-0.47
2
Ext.
0
0
0
-2
-2
-2
-4
-4
-4
-4
-4
-4
4
-2
0
2
4
Flex.
Flex.
-2
0
2
4
.
-2
0
2
MUS [N m]
Ext.
INT [N m]
Elbow
20
.
-0.19
20
-0.20
20
10
10
10
0
0
0
-10
-10
-10
-20
-20
-20
-20
-20
0
20
Flex.
Flex.
Ext.
Ext.
-20
0
20
.
0
20
MUS [N m]
-0.22
Wrist. For all speeds, before the movements, the wrist flexion muscle torque was kept at a magnitude that opposed the
extension gravity torque and it held the wrist in the starting
position (Fig. 2D). After the start of the movement, the wrist
muscle torque mirrored and counteracted the interaction torque
throughout the movement. It should be noted that the wrist
flexion muscle torque was appropriately scaled to counteract
the large wrist extension interaction torque in the fast condition.
The wrist interaction torque was also plotted in Fig. 3A
against the wrist muscle torque from the movement onset to the
ball-release time for the same trial of Fig. 2. In Fig. 3, A and B,
at the wrist, diagonal straight lines are present in all conditions.
A diagonal straight line indicates that the interaction torque and
the muscle torque are approximately mirror images of each
other (Bastian et al. 1996). All IOCIMs in Fig. 3, A and B, for
the wrist showed negative sign (i.e., the counteractive relationship).
THREE-JOINT THROWING. Figure 4A shows stick pictures of one
trial for subject A, when throwing balls at three different speeds
(slow, medium, and fast) with three joints. Figure 4, B and C,
shows the kinematics of the same trials of Fig. 4A. Figure 4D
shows the kinetics of them. The ball-release time was set to 0
J Neurophysiol • VOL
ms. At first the shoulder started to extend (slow, ⫺360 ms;
medium, ⫺265 ms; fast, ⫺260 ms; see Fig. 4C). Next the
elbow started to extend. Finally the wrist started to flex, and the
ball was released (0 ms).
Shoulder. The kinetic feature of the shoulder during threejoint throwing was similar to that of the elbow during two-joint
throwing. There was an initial phase in which the effect of the
interaction torque was little. In the fast condition, for example,
the initial phase was from about ⫺250 to about ⫺100 ms. In
this phase, the subject increased the shoulder extension muscle
torque to produce the shoulder extension net torque, which initiated the three-joint throwing (Fig. 4, C and D). Next the shoulder
flexion interaction torque had very large effect on the shoulder net
torque; as a consequence, the shoulder extension net toque was
changed to the shoulder flexion net torque (Fig. 4D).
The shoulder interaction torque was plotted in Fig. 5A
against the shoulder muscle torque from the movement onset to
the ball-release time for the same trial of Fig. 4. In the shoulder
joint in Fig. 5A, most of the plots exist in the second quadrant.
This means that the interaction torque and the muscle torque
always showed different signs, i.e., the counteractive relationship. All IOCIMs for the shoulder show negative sign (i.e., the
counteractive relationship). Figure 5B shows the interaction
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-20
-0.16
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
Slow
A
Medium
1789
Fast
Ball
Angle [deg]
Anglular Velocity
[deg. s-1]
Shoulder
Extension
Elbow
Extension
Wrist
Flexion
200
180
160
140
120
100
80
60
40
20
0
-20
0
0
-500
-500
-1000
-1000
0
Shoulder
-500
Extension
Elbow
Extension
-1000
Wrist
Flexion
Wrist
Torque [N m]
-200
D
20 cm
200
180
160
140
120
100
80
60
40
20
0
-20
-100
0 [ms]
-200
-100
0 [ms]
2
2
0
0
-2
-2
10
10
0
0
-10
-10
-20
-20
50
50
25
Flexion 25
25
0
0
0
-25
-25
2
Extension
-200
-100
0 [ms]
-200
-100
0 [ms]
.
0
Flexion
-2
Elbow
Torque [N m]
10
.
0
50
Shoulder
Torque [N m]
Extension
-10
-20
.
Flexion
Net
Gravity
Muscle
Interaction
Extension
-25
-200
-100
0 [ms]
-200
-100
0 [ms]
FIG.
4. Data for representative single trials in 3-joint throwing for slow (left), medium (middle), and fast condition (right) from
subject A are shown. A: stick pictures at ⫺100, ⫺75, ⫺50, ⫺25, and 0 ms are drawn. The target is in the right direction. B: joint
angles at the wrist (dashed line) and elbow (solid line), and shoulder (thick line). C: joint angular velocity at the wrist (dashed line)
elbow (solid line), and shoulder (thick line). D: the net torque (solid line), gravity torque (dashed line), muscle torque (gray line),
and interaction torque (dotted line) at each joint. The vertical lines at 0 ms represent the ball-release time. Four vertical dotted lines
represent ⫺100, ⫺75, ⫺50, and ⫺25 ms.
torques of all the trials for all the subjects plotted against their
muscle torques for three-joint throwing. Averages of IOCIM
across all the subjects are also shown. They all show negative
sign at the shoulder (i.e., the counteractive relationship).
Elbow. The kinetic feature of the elbow during three-joint
throwing was clearly different from that of the elbow during
J Neurophysiol • VOL
two-joint throwing. The elbow extension interaction torque
assisted the elbow extension muscle torque for all conditions, though in the slow and medium condition, the assistive interaction torque is small throughout the throwing
(Fig. 4D). It should be noted that, in the fast condition, large
elbow extension net torque was generated by the large
89 • APRIL 2003 •
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C
Ball
Release
Elbow
Ball
Shoulder
Release
200
180
160
140
120
100
80
60
40
20
0
-20
B
20 cm
MP
Wrist
1790
M. HIRASHIMA, K. KUDO, AND T. OHTSUKI
A
Slow
Medium
4
4
INT [N m]
Wrist
-0.39
.
Fast
4
-0.25
-0.35
2
2
2
0
0
0
-2
-2
-2
-4
-4
-4
-4
-4
-4
4
-2
0
2
4
Ext.
Flex.
Flex.
-2
0
2
4
.
-2
0
2
MUS [N m]
Ext.
INT [N m]
Elbow
20
.
20
+0.05
20
+0.10
10
10
10
0
0
0
-10
-10
-10
-20
-20
-20
-20
0
20
Flex.
Ext.
Ext.
-20
0
20
.
0
20
MUS [N m]
Flex.
INT [N m]
Shoulder
100
.
100
-0.93
50
100
-0.82
50
0
0
0
-50
-50
-50
-100
-100
Flex.
Ext.
Ext.
-100
100 -100
0
Flex.
0
FIG. 5. A: interaction torque plotted
against the muscle torque from movement
onset to ball release of the same trial as
Fig. 4. The IOCIMs are also shown. B:
interaction torque plotted against the muscle torque from movement onset to ball
release of all trials for all subjects in 3-joint
throwing. The IOCIMs averaged for all
subjects are also shown.
100
.
0
MUS [N m]
-100
100 -100
-0.94
50
B
Slow
Medium
4
4
INT [N m]
Wrist
-0.28
.
Fast
4
-0.23
-0.35
2
2
2
0
0
0
-2
-2
-2
-4
-4
-4
-4
-4
-4
4
-2
0
2
4
Ext.
Flex.
Flex.
-2
0
2
4
.
-2
0
2
MUS [N m]
Ext.
INT [N m]
Elbow
20
.
+0.06
20
+0.14
20
10
10
10
0
0
0
-10
-10
-10
-20
-20
-20
-20
-20
0
20
Flex.
Flex.
Ext.
Ext.
-20
0
20
.
0
20
MUS [N m]
+0.25
100
100
-0.63
INT [N m]
Shoulder
100
.
-0.73
-0.87
50
50
50
0
0
0
-50
-50
-50
-100
-100
0
Flex.
Ext.
Ext.
-100
100 -100
0
100
.
0
MUS [N m]
-100
100 -100
Flex.
extension interaction torque assisting the extension muscle
torque (Fig. 4D), and as a consequence, large elbow angular
velocity for extension occurred (Fig. 4C). Figure 5, A and B,
shows that most of the plots for the elbow are in the third
quadrant and that all IOCIM values show positive signs (i.e.,
J Neurophysiol • VOL
the assistive relationship), though in the slow condition the
IOCIM is almost zero.
Wrist. The kinetic feature of the wrist during three-joint
throwing was very similar to that of the wrist during two-joint
throwing. The wrist muscle torque mirrored and counteracted
89 • APRIL 2003 •
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-20
+0.25
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
Strategy to adjust the ball speed
The intersegmental dynamics is greatly influenced by the
joint angular velocity and angular acceleration (Hollerbach and
Flash 1982). Therefore throwers must generate an appropriate
motor command by anticipating the complex and time-varying
interaction torque caused by the throwing motion itself with the
desired speed. It is totally unknown how the CNS deals with
the interaction torque when adjusting the movement speed in
ball-throwing movements. Is there any effective strategy to
ensure the accuracy of the performance when adjusting the ball
speed from throw to throw?
At first, it was examined whether subjects could throw balls
with three different speeds. Repeated-measures ANOVAs revealed a significant main effect of speed specification [2-joint
throwing: F(2,14) ⫽ 83.0, 3-joint throwing: F(2,12) ⫽73.6] on
the actual ball speed. Tukey’s post hoc multiple comparison
tests indicated significant differences for all comparisons
among slow [5.46 ⫾ 0.30 (SD) m/s], medium (6.49 ⫾ 0.41
m/s), and fast (8.02 ⫾ 0.51 m/s) conditions in two-joint throwing (P ⬍ 0.05) and for all comparisons among slow (6.07 ⫾
0.32 m/s), medium (8.27 ⫾ 0.54 m/s), and fast (11.4 ⫾ 1.33
m/s) conditions in three-joint throwing (P ⬍ 0.05). Subjects
could throw the balls using three different speeds. In addition,
it was examined whether angular velocity for each joint at the
ball-release time changed with the three speed conditions.
TWO-JOINT THROWING. The angular velocity at the ball-release
time during two-joint throwing, which was averaged across all
subjects, is shown in Fig. 6, right. Repeated-measures
ANOVAs were performed to assess the effect of speed on
angular velocity at the ball-release time at the elbow and wrist
joints. It revealed a significant main effect of speed at the
elbow [F(2,14) ⫽ 82.6]. However, interestingly, there was no
significant main effect of speed at the wrist [F(2,14) ⫽ 0.18;
P ⬎ 0.8). The results of Tukey’s post hoc multiple comparison
tests are shown in Fig. 6, right. This indicates that the wrist
angular velocity at the ball-release time was kept relatively constant in spite of the increase of ball speed and that only elbow
angular velocity contributed to the increase of ball speed.
We examined the mechanism whereby the wrist angular
velocity at the ball-release time was kept constant by calculating the muscle torque impulse (MUSIm), the interaction torque
impulse (INTIm), and the net torque impulse (NETIm), which
are calculated as follows
MUSIm ⫽
冕
冕
冕
T2
MUS共t兲dt
T1
INTIm ⫽
T2
INT共t兲dt
T1
NETIm ⫽
T2
NET共t兲dt
T1
where T1 is the time of movement onset at the most proximal
joint and T2 is the ball-release time. MUSIm, INTIm, and
FIG. 6. Muscle torque impulse (MUSIm),
interaction torque impulse (INTIm), net
torque impulse (NETIm) from movement onset to ball release, and angular velocity at ball
release for each joint in 2-joint throwing. S,
slow; M, medium; F, fast. *P ⬍ 0.05
(Tukey’s post hoc multiple comparison test).
J Neurophysiol • VOL
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the interaction torque to allow a controlled wrist flexion (Fig.
4D). In Fig. 5, A and B, diagonal straight lines were clear for
all conditions at the wrist. All IOCIM values showed negative
signs (i.e., the counteractive relationship).
SUMMARY. The general features of intersegmental dynamics
for the planar throwing movements can be summarized as
follows. 1) At the proximal joint of the most proximal segment
(i.e., the elbow joint in 2-joint throwing and the shoulder joint
in 3-joint throwing), there was an initial phase in which the
effect of the interaction torque was little, in other words the
change of the net torque was mainly determined by the change
of the muscle torque. This net torque initiated the throwing
motion. Next, the interaction torque occurred for the opposite
direction to the muscle torque. 2) At the wrist joint, the
counteractive relationship between muscle torque and interaction torque was always observed. Interestingly, the larger wrist
flexion muscle torque counteracted the larger wrist extension
interaction torque. 3) The difference of the intersegmental
dynamics, which was caused by the number of the mobilized
joints, was seen at the elbow especially in the fast condition. In
three-joint throwing, the elbow extension interaction torque
assisted the elbow extension muscle torque to generate the
large elbow extension net torque. This assistive relationship
contributed to generating very large angular velocity at the
elbow for the fast condition.
1791
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M. HIRASHIMA, K. KUDO, AND T. OHTSUKI
MUSIm
Wrist
[N.m.ms]
250
200
150
100
50
0
-50
-100
-150
-200
-250
1500
Elbow
1000
500
0
-500
[N.m.ms]
100
50
0
-50
-100
600
400
200
0
-200
-1500
-400
-600
4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-1000
Shoulder
INTIm
NETIm
[N.m.ms]
80
60
40
20
0
-20
-40
-60
-80
1500
1000
500
0
-500
-1000
-1500
1000
500
0
-500
-1000
J Neurophysiol • VOL
significant main effect of speed at the wrist [F(2,12) ⫽ 0.09;
P ⬎ 0.9] as well as for two-joint throwing. The results of
Tukey’s post hoc multiple comparison tests again testified that
wrist angular velocity at the ball-release time was kept relatively constant in spite of the increase of ball speed and that
only the shoulder and elbow angular velocity contributed to the
increase of ball speed.
Repeated-measures ANOVAs on the MUSIm, INTIm, and
NETIm at the shoulder, elbow, and wrist joints revealed significant main effects of speed on the shoulder MUSIm
[F(2,12) ⫽ 90.0], the shoulder INTIm [F(2,12) ⫽ 42.7], the
shoulder NETIm [F(2,12) ⫽ 4.26], the elbow MUSIm
[F(2,12) ⫽ 12.3], the elbow INTIm [F(2,12) ⫽ 15.5], the
elbow NETIm [F(2,12) ⫽ 40.7], the wrist MUSIm [F(2,12) ⫽
14.1], and the wrist INTIm [F(2,12) ⫽ 23.3] but no significant
main effect of speed on the wrist NETIm [F(2,12) ⫽ 0.17; P ⬎
0.8]. The results of Tukey’s post hoc multiple comparison tests
are shown in Fig. 7.
Figure 7 indicates that the method for increasing the shoulder extension angular velocity at the ball release during threejoint throwing was the same as that for increasing the elbow
extension angular velocity during two-joint throwing, i.e., the
CNS increased the shoulder extension MUSIm that overcame
the increase of the shoulder flexion INTIm.
At the elbow, the MUSIm’s contribution to the increase of
the NETIm was not particularly large as shown by the fact that
there was no significant difference in the MUSIm between the
slow and fast conditions. Alternatively, the INTIm in the fast
condition was significantly larger than that in the slow condition. As the direction of the elbow INTIm in the three-joint
throwing was extension (desired direction), the CNS could
Angular velocity
at ball release
[deg. s-1]
800
600
400
200
0
-200
-400
-600
-800
1000
Extension
Flexion
Flexion
500
0
-500
-1000
400
300
200
100
0
-100
-200
-300
-400
89 • APRIL 2003 •
Extension
Flexion
Extension
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FIG. 7. MUSIm, INTIm, NETIm from
movement onset to ball release, and angular
velocity at ball release for each joint in
3-joint throwing. *P ⬍ 0.05 (Tukey’s post
hoc multiple comparison test).
Downloaded from http://jn.physiology.org/ by 10.220.33.3 on July 28, 2017
NETIm, which were averaged across all subjects, are shown in
Fig. 6. Repeated-measures ANOVAs were performed to assess
the effect of speed on the MUSIm, INTIm, and NETIm at the
elbow and wrist joints, revealing significant main effects of
speed on the elbow MUSIm [F(2,14) ⫽ 75.8], the elbow
INTIm [F(2,14) ⫽ 5.43], the elbow NETIm [F(2,14) ⫽ 54.4],
the wrist MUSIm [F(2,14) ⫽ 67.7], and the wrist INTIm
[F(2,14) ⫽ 39.3] but no significant main effect of speed on the
wrist NETIm [F(2,14) ⫽ 2.85; P ⬎ 0.09]. The results of
Tukey’s post hoc multiple comparison tests on the MUSIm,
INTIm, and NETIm in two-joint throwing are shown in Fig. 6.
Figure 6 shows that, although the elbow flexion INTIm
increased a little with the ball speed, the increase of the elbow
extension MUSIm overcame it to realize the increase of the
elbow extension angular velocity with the ball speed. In contrast, the wrist extension INTIm also increased with the ball
speed, and this increase of the wrist extension INTIm was
exactly counteracted by the increase of the wrist flexion
MUSIm; as a result, the wrist flexion angular velocity at the
ball-release time was kept relatively constant irrespective of
the ball speed. In addition, interestingly, a time-series of the
wrist angular velocity between ⫺80 and 0 ms was also kept
relatively constant irrespective of the ball speed (see Fig. 2C).
THREE-JOINT THROWING. The same analysis was also performed for three-joint throwing. The angular velocity at the
ball-release time during three-joint throwing, which was averaged across all subjects, is shown in Fig. 7, right. Repeatedmeasures ANOVAs were performed to assess the effect of
speed on angular velocity at the ball-release time at the shoulder, elbow, and wrist joints. They revealed significant main
effects of speed at the shoulder [F(2,12) ⫽ 21.8] and at the
elbow [F(2,12) ⫽ 46.7]. However, interestingly, there was no
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
utilize this elbow extension INTIm to increase the elbow
extension angular velocity at the ball-release time.
In the same way as in the two-joint throwing, the increase of
the wrist extension INTIm was exactly counteracted by the
increase of the wrist flexion MUSIm in the three-joint throwing. Thus the wrist flexion angular velocity at the ball-release
time was kept relatively constant irrespective of the ball speed.
In addition, the time series of the wrist angular velocity between ⫺80 and 0 ms was also kept relatively constant irrespective of the ball speed (see Fig. 4C).
DISCUSSION
Hierarchical control
Assistive relationship between interaction and muscle torque
Putnam (1991, 1993) provided insights into the intersegmental dynamics in walking, ball kicking, and ball throwing. Putnam’s discussion was similar to hierarchical control theory,
which suggests that the interaction torque produced by the
shoulder motion was utilized at the elbow in ball throwing. In
this study, the existence of the assistive relationship between
interaction and muscle torque was confirmed at the elbow in
three-joint throwing. However, the assistive relationship was
not identified at the wrist. Interestingly, Koshland et al. (2000)
reported that no case of reaching has been described in which
J Neurophysiol • VOL
the wrist muscle torque assisted the interaction torques at the
wrist joint, giving several examples (Dounskaia et al. 1998;
Ghez et al. 1996; Virji-Babul and Cooke 1995). It remains to
be answered whether the counteractive relationship at the wrist
is caused by the neural contribution or the musculoskeletal
properties at the wrist or a combination of these two factors.
Compensation for the interaction torque at the wrist
At the wrist joint, the muscle torque exactly counteracted the
interaction torque during both two- and three-joint throwing.
The mirror relationship between the interaction and muscle
torque indicates that this compensation would be performed by
the feedforward control. On-line feedback is not feasible. If
compensation was performed by on-line feedback, a corrective
motor command would no longer be appropriate for the current
state of the limb due to the feedback delay, especially in a fast
movement, such as ball throwing.
Sainburg et al. (1993, 1995) supposed that the deficit in
interjoint coordination of the patients without proprioception
was not attributed to deficits in on-line feedback. They hypothesized that proprioceptive information is used to update the
internal model of the limb in the current environment. Ghez et
al. (1990) and Godon et al. (1995) demonstrated that visual
information of a limb movement could improve interjoint
coordination of patients without proprioception. On the other
hand, visual information could not improve the interjoint coordination of cerebellar patients (Beppu et al. 1987; Brown et
al. 1993; van Donkelaar and Lee 1994). This indicates that the
cerebellum is responsible for updating the internal model by
using proprioceptive or visual information. By using an fMRI,
Imamizu et al. (2000) showed that the cerebellum is involved
in acquiring an internal model.
These findings are compatible with the studies about ball
throwing by Hore and Timmann. Skilled throwers can predict
the back force from the ball and compensate for it by producing
appropriate finger grip force (Hore et al. 1999, 2001). On the
other hand, cerebellar patients and unskilled subjects could not
control finger grip force appropriately (Timmann et al. 2001).
These findings indicate that poor throwers have not acquired
the appropriate internal model for throwing. Results of our
present study further indicate that throwers can also predict and
compensate for the interaction torque at the wrist.
However, it is noted that the muscle torque calculated by
inverse dynamics includes not only the mechanical contribution of active muscle contraction at the joint but also passive
contributions (passive torque) by muscles, tendons, ligaments,
articular capsules, and other connective tissues. Therefore the
possibility that the compensation for the interaction torque was
achieved by the passive torque at the wrist joint cannot be
excluded. Furthermore, the passive torque appears with a feedback delay of 0 ms. As noted by Dounskaia et al. (1998),
attention must be given to the passive torque, if the joint rotates
around the extremity of the range of motion. Stiffness control
is also feasible. Osu et al. (2002) hypothesized that viscoelasticity contributes more when an internal model is inaccurate,
while internal models contribute more after the accurate internal model is acquired.
To determine whether this compensation is realized by the
feedforward time-varying control or the stiffness control, the
electromyographic (EMG) studies recording the agonists and an-
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The results showed that, at the most proximal joint (i.e., the
elbow joint in 2-joint throwing and the shoulder joint in 3-joint
throwing), the muscle torque counteracted the interaction
torque as well as at the wrist joint, which is indicated by the
fact that the IOCIM values showed negative signs. However,
the role of the muscle torque at the most proximal joint is
different from that at the wrist. At the most proximal joint,
there was an initial phase in which the change of the muscle
torque predominantly determined the change of the net torque.
This initial net torque initiated the throwing motion. The role
of the muscle torque in the initial phase was not counteracting
the interaction torque but accelerating the most proximal segment. Furthermore, in the three-joint throwing, the role of the
shoulder muscle torque in the initial phase was not only to
accelerate the shoulder joint but also to produce the assistive
interaction torque for the elbow.
Dounskaia et al. (1998) suggested a hierarchical control, in
which the role of the proximal muscle is to generate movement of
the whole linkage and the role of the distal muscle is to produce
corrections of the movement necessary to fulfill the task. Bernstein (1967, 1996) insightfully suggested that the role of the
muscle activity is not only to accelerate the limb but also to
control the intersegmental interaction. The results here support
these ideas. In three-joint throwing, throwers effectively utilized
assistive interaction torque and produced larger elbow angular
velocity to satisfy the demand of the fast condition. It is also
interesting that the CNS realized the increase of the elbow angular
velocity depending mainly on the increase of the assistive interaction torque impulse (INTIm) in three-joint throwing (Fig. 7) but
on the muscle torque impulse (MUSIm) in two-joint throwing
(Fig. 6). The role of the elbow muscle torque in three-joint
throwing was to assist the effects of intersegmental interaction,
whereas the role of the elbow muscle torque in two-joint throwing
was to accelerate the limb by itself.
1793
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M. HIRASHIMA, K. KUDO, AND T. OHTSUKI
tagonists at the wrist are necessary. If there was co-activation of
agonists and antagonists, it would be likely that the CNS used the
stiffness control. However, some recent studies revealed the reciprocal EMG pattern between the flexor (agonist) and extensor
(antagonist) of the wrist during overarm badminton smash stroke
(Sakurai and Ohtsuki 2000) and baseball pitching (Hirashima et
al. 2002), implying feedforward time-varying control. The computer simulation would be also useful to determine whether it is
possible or not for the high stiffness at the wrist to realize this
compensation. At the present stage, it appears that throwers can
compensate for the interaction torque at the wrist by the neuromusculoskeletal system as a whole.
⫹ ␪˙ 1␪˙ 2关2m2l1r2 sin ␪2兴
⫺ ẍ e关共m2l1 ⫹ m1r1兲 cos ␪1 ⫹ m2r2 cos共␪1 ⫹ ␪2兲兴
⫺ ÿ e关共m2l1 ⫹ m1r1兲 sin ␪1 ⫹ m2r2 sin共␪1 ⫹ ␪2兲兴
GRA e ⫽ ⫺ g关共m2l1 ⫹ m1r1兲 sin ␪1 ⫹ m2r2 sin共␪1 ⫹ ␪2兲兴
MUS e ⫽ NET e ⫺ INT e ⫺ GRA e
WRIST
NET w ⫽ ⫹ ␪¨ 2关I2 ⫹ m2r22兴
INT w ⫽ ⫺ ␪¨ 1关I2 ⫹ m2r22 ⫹ m2l1r2 cos ␪2兴
⫺ ␪˙ 21关m2l1r2 sin ␪2兴
⫺ ẍ e关m2r2 cos共␪1 ⫹ ␪2兲兴
The role of the wrist in ball throwing is discussed here. The
strategy used by the CNS was that the elbow and shoulder
contributed to adjustments of ball speed but the wrist did not.
As the length of the hand is shorter than that of the forearm and
upper arm, the angular velocity of the wrist is less effective for
increasing ball speed than the angular velocity of the elbow and
shoulder. This may be one of the reasons why the wrist does
not contribute to the adjustment of ball speed. However, a more
feasible reason would be that the role of the wrist is to simplify
the control of the finger grip force, leading to an accurate ball
release. Because the extrinsic finger muscles cross the wrist
joint, the wrist motion affects the length and velocity of the
extrinsic finger muscles and finally affects the force-producing
capacity of the extrinsic finger muscles (Brand 1985; Herrmann and Delp 1999; O’Driscoll et al. 1992). Werremeyer and
Cole (1997) showed that the wrist motion affects the precision
of the grip force. Therefore the fact that there was an insignificant difference in the wrist angular velocity at ball release at
three speeds ensures the same force-producing capacity of the
extrinsic finger muscles at ball release irrespective of the ball
speed. In addition, the time series of the wrist angular velocity
between ⫺80 and 0 ms was also kept relatively constant
irrespective of the ball speed.
By virtue of this wrist kinematics, the extrinsic finger muscles can produce the finger grip force following the same time
schedule irrespective of the ball speed even under the circumstances that the finger grip force must be scaled to a back force
from the ball that is proportional to the hand acceleration (Hore
et al. 1999, 2001). This would reduce the CNS’s load of having
to change the finger control programs from throw to throw. To
confirm this hypothesis, it would be necessary to show some
correlation between the target hitting performance and the
wrist kinematic variability. The present study does not show
this correlation, probably because the throwing task was relatively easy. A throwing task that requires more accurate finger
control would be necessary.
⫺ ÿ e关m2r2 sin共␪1 ⫹ ␪2兲兴
GRA w ⫽ ⫺ g关m2r2 sin共␪1 ⫹ ␪2兲兴
MUS w ⫽ NET w ⫺ INT w ⫺ GRA w
Ii ⫽ moment of inertia about the center of gravity, ri ⫽
distance to center of mass from proximal joint of the segment, li ⫽ length,
mi ⫽ mass (i⫽ 1: forearm, 2: hand), ẍe ⫽ linear acceleration of the elbow
for horizontal direction, ÿe⫽ linear acceleration of the elbow for vertical
direction. The mass of the ball is included in the hand mass.
SYMBOLS.
Three-joint throwing
SHOULDER
¨ 1关I1 ⫹ I2 ⫹ I3 ⫹ m2l21 ⫹ m3l21 ⫹ m3l22 ⫹ m1r21 ⫹ m2r22 ⫹ m3r23
NET s ⫽ ⫹ ␾
⫹ 共2m 3l1l2 ⫹ 2m2l1r2兲 cos ␾2 ⫹ 2m3l2r3 cos ␾3
⫹ 2m 3l1r3 cos共␾2 ⫹ ␾3兲
¨ 2关I2 ⫹ I3 ⫹ m3l22 ⫹ m2r22 ⫹ m3r23 ⫹ 共m3l1l2 ⫹ m2l1r2兲 cos ␾2
INT s ⫽ ⫺ ␾
⫹ 2m3l2r3 cos ␾3 ⫹ m3l1r3 cos共␾2 ⫹ ␾3兲兴
¨ 3关I3 ⫹ m r ⫹ m3l2r3 cos ␾3 ⫹ m3l1r3 cos共␾2 ⫹ ␾3兲兴
⫺␾
2
3 3
˙ 22关l1共共m3l2 ⫹ m2r2兲 sin ␾2 ⫹ m3r3 sin共␾2 ⫹ ␾3兲兲兴
⫹␾
˙ 23关m3l2r3 sin ␾3 ⫹ m3l1r3 sin共␾2 ⫹ ␾3兲兴
⫹␾
˙ 1␾
˙ 2关2l1共共m3l2 ⫹ m2r2兲 sin ␾2 ⫹ m3r3 sin共␾2 ⫹ ␾3兲兲兴
⫹␾
˙ 1␾
˙ 3关2m3r3共l2 sin ␾3 ⫹ l1 sin共␾2 ⫹ ␾3兲兲兴
⫹␾
˙ 3关2m3r3共l2 sin ␾3 ⫹ l1 sin共␾2 ⫹ ␾3兲兲兴
˙ 2␾
⫹␾
⫺ ẍ s关共m2l1 ⫹ m3l1 ⫹ m1r1兲 cos ␾1 ⫹ 共m3l2 ⫹ m2r2兲 cos共␾1 ⫹ ␾2兲
⫹ m3r3 cos共␾1 ⫹ ␾2 ⫹ ␾3兲兴
⫺ ÿ s关共m2l1 ⫹ m3l1 ⫹ m1r1兲 sin ␾1 ⫹ 共m3l2 ⫹ m2r2兲 sin共␾1 ⫹ ␾2兲
⫹ m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
GRA s ⫽ ⫺ g关共m2l1 ⫹ m3l1 ⫹ m1r1兲 sin ␾1 ⫹ 共m3l2 ⫹ m2r2兲 sin共␾1 ⫹ ␾2兲
APPENDIX
⫹ m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
Two-joint throwing
MUS s ⫽ NET s ⫺ INT s ⫺ GRA s
ELBOW
ELBOW
NET e ⫽ ⫹ ␪¨ 1关I1 ⫹ I2 ⫹ m2l21 ⫹ m1r21 ⫹ m2r22 ⫹ 2m2l1r2 cos ␪2兴
¨ 2关I2 ⫹ I3 ⫹ m3l22 ⫹ m2r22 ⫹ m3r23 ⫹ 2m3l2r3 cos ␾3兴
NET e ⫽ ⫹ ␾
INT e ⫽ ⫺ ␪¨ 2关I2 ⫹ m2r22 ⫹ m2l1r2 cos ␪2兴
¨ 1关I2 ⫹ I3 ⫹ m3l22 ⫹ m2r22 ⫹ m3r23 ⫹ 共m3l1l2
INT e ⫽ ⫺ ␾
⫹ ␪˙ 22关m2l1r2 sin ␪2兴
⫹ m2l1r2兲 cos ␾2 ⫹ 2m3l2r3 cos ␾3 ⫹ m3l1r3 cos共␾2 ⫹ ␾3兲兴
J Neurophysiol • VOL
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Role of the wrist
UTILIZATION AND COMPENSATION OF INTERACTION TORQUES IN THROWING
¨ 3关I3 ⫹ m3r23 ⫹ m3l2r3 cos ␾3兴
⫺␾
˙ 21关l1共共m3l2 ⫹ m2r2兲 sin ␾2 ⫹ m3r3 sin共␾2 ⫹ ␾3兲兲兴
⫺␾
˙ 23关m3l2r3 sin ␾3兴
⫹␾
˙ 1␾
˙ 3关2m3l2r3 sin ␾3兴
⫹␾
˙ 2␾
˙ 3关2m3l2r3 sin ␾3兴
⫹␾
⫺ ẍ s关共m3l2 ⫹ m2r2兲 cos共␾1 ⫹ ␾2兲 ⫹ m3r3 cos共␾1 ⫹ ␾2 ⫹ ␾3兲兴
⫺ ÿ s关共m3l2 ⫹ m2r2兲 sin共␾1 ⫹ ␾2兲 ⫹ m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
GRA e ⫽ ⫺ g关共m3l2 ⫹ m2r2兲 sin共␾1 ⫹ ␾2兲 ⫹ m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
MUS e ⫽ NET e ⫺ INT e ⫺ GRA e
WRIST
¨ 3关I3 ⫹ m3r23兴
NET w ⫽ ⫹ ␾
¨ 2关I3 ⫹ m3r23 ⫹ m3l2r3 cos ␾3兴
⫺␾
˙ 21关m3r3共l2 sin ␾3 ⫹ l1 sin共␾2 ⫹ ␾3兲兲兴
⫺␾
˙ 22关m3l2r3 sin ␾3兴
⫺␾
˙ 1␾
˙ 2关2m3l2r3 sin ␾3兴
⫺␾
⫺ ẍ s关m3r3 cos共␾1 ⫹ ␾2 ⫹ ␾3兲兴
⫺ ÿ s关m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
GRA w ⫽ ⫺ g关m3r3 sin共␾1 ⫹ ␾2 ⫹ ␾3兲兴
MUS w ⫽ NET w ⫺ INT w ⫺ GRA w
Ii ⫽ moment of inertia about the center of gravity, ri ⫽
distance to center of mass from proximal joint of the segment, li ⫽
length, mi ⫽ mass (i ⫽ 1: upper arm, 2: forearm, 3: hand), ẍs ⫽ linear
acceleration of the shoulder for horizontal direction, ÿs ⫽ linear
acceleration of the shoulder for vertical direction. The mass of the ball
is included in the hand mass.
SYMBOLS.
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