J Neurophysiol 90: 1449 –1463, 2003;
10.1152/jn.00220.2003.
Counteractive Relationship Between the Interaction Torque and Muscle
Torque at the Wrist Is Predestined in Ball-Throwing
Masaya Hirashima,1 Kunishige Ohgane,2 Kazutoshi Kudo,1 Kazunori Hase,2 and Tatsuyuki Ohtsuki1
1
Department of Life Sciences (Sports Sciences), Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902;
and 2National Institute of Advanced Industrial Science and Technology, Tsukuba Central 6, Ibaraki 305-8566, Japan
Submitted 10 March 2003; accepted in final form 16 May 2003
INTRODUCTION
Single-joint movements can be mechanically explained only
by the muscle torque and the gravity torque acting at the joint.
However, the multi-joint movement is not very simple because
torque at one joint includes not only the muscle torque and the
gravity torque but also interaction torques due to the rotations
of the other joints (Hasan 1991; Hollerbach and Flash 1982;
Smith and Zernicke 1987). The interaction torque is a load that
is self-generated by the movement itself and is absent before
the movement begins. Therefore to perform the desired multijoint movement, the CNS should predict the interaction torque
and produce the appropriate motor command in advance by
taking into account the interaction torque, whether the interacAddress for reprint requests: M. Hirashima, 噦 Dr. T. 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]).
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tion torque is counteractive or assistive to the muscle torque
(Bastian et al. 1996, 2000; Beer et al. 2000; Dounskaia et al.
2002a,b; Galloway and Koshland 2002; Gribble and Ostry
1999; Koshland et al. 2000; Pigeon et al. 2003; Sainburg et al.
1993, 1995, 1999; Topka et al. 1998). Indeed, cerebellar patients have shown their inability of dealing with the interaction
torque and an abnormally curved hand path was produced
(Bastian et al. 1996). Deafferented patients have also shown
this deficit (Ghez and Sainburg 1995; Sainburg et al. 1993,
1995).
In this study, we focused on swing motions such as ballthrowing, ball-kicking, and the swing phase of locomotion in
which the motion of the proximal joint (shoulder, hip) produced assistive interaction torque for the distal joint (elbow,
knee). This assistive interaction torque plays an important role
for forming these swing motions (Feltner 1989; Hoy and Zernicke 1985; Mena et al. 1981; Phillips et al. 1983; Putnam
1991, 1993). Recently, we examined the relationship between
the interaction torque and muscle torque at the joints of the
upper extremities during ball-throwing and confirmed that the
shoulder muscle torque produced the assistive interaction
torque for the elbow, which was effectively utilized to generate
large elbow angular velocity when throwing fast (Hirashima et
al. 2003). In line with these studies, the shoulder and elbow
motions would be expected to produce the assistive interaction
torque for the wrist joint as well. However, this was not the
case. In a recent experiment, we revealed that the interaction
torque at the wrist was always counteractive to the muscle
torque (Hirashima et al. 2003). It would be interesting to
determine whether the interaction torque at the wrist joint can
be assistive in ball-throwing. The purpose of this study is to
clarify, by means of a computer simulation of the two-joint
throwing (elbow and wrist), whether the counteractive relationship at the wrist during ball-throwing is caused by the neural
contribution or the musculoskeletal mechanical properties at
the wrist or a combination of these two factors.
It is very important for investigators to know the dynamical
aspects of the limb movements that are determined by the
mechanical property of the human arm to understand the
CNS’s ability to perform the desired multi-joint limb movements. Although the interaction torque counteractive to the
muscle torque is mechanically disadvantageous for generating
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0022-3077/03 $5.00 Copyright © 2003 The American Physiological Society
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Hirashima, Masaya, Kunishige Ohgane, Kazutoshi Kudo, Kazunori Hase, and Tatsuyuki Ohtsuki. Counteractive relationship
between the interaction torque and muscle torque at the wrist is
predestined in ball-throwing. J Neurophysiol 90: 1449 –1463, 2003;
10.1152/jn.00220.2003. Many investigators have demonstrated that in
swing motions such as ball-throwing, the motion of the proximal joint
(shoulder) produced assistive interaction torque for the distal joint
(elbow). In line with these studies, the shoulder and elbow motions
would be expected to produce the assistive interaction torque for the
wrist joint as well. However, we recently showed that the interaction
torque at the wrist was always counteractive to the wrist muscle
torque during ball-throwing. The purpose of this study is to clarify, by
means of computer simulation, whether the counteractive relationship
at the wrist during ball-throwing is caused by the neural contribution
or the musculoskeletal mechanical properties of the human arm. First,
we simulated the throwing motions of the normal forearm-hand model
by systematically changing the proximal-to-distal delay of muscle
activities and could line up two candidates for the determinant of the
counteractive relationship: the rest angle (neutral angle) of the wrist
and the length and mass of the hand. Second, we simulated the
throwing motions of the virtual forearm-hand models, showing that
only nonrealistic elongation of these two parameters produced the
assistive relationship between the interaction torque and muscle
torque. These results suggested that the mechanical properties of the
human wrist are the main determinant of the counteractive relationship, which is advantageous for keeping the state of the wrist joint
stable in multi-joint upper-limb movements and would lead to avoidance of excessive wrist extension or flexion and simplification of
extrinsic finger control.
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HIRASHIMA, OHGANE, KUDO, HASE, AND OHTSUKI
large angular velocity at the joint, it may have some alternative
merits. We will discuss some feasible merits of the counteractive relationship at the wrist joint, relating the wrist motion
with the extrinsic finger muscle control. Finally, we will discuss the CNS’s ability to deal with the mechanical properties of
the body, showing the separate roles of proximal and distal
joints in ball-throwing.
METHODS
Model
M共␪兲␪¨ ⫽ 关T ⫺ V共␪, ␪˙ 兲 ⫺ G共␪兲兴
(1)
where ␪ is the vector of the joint angle, ␪˙ is the vector of joint angular
velocity, ␪¨ is the vector of joint angular acceleration, T is the vector
of the torque, M(␪) is the inertia matrix, V(␪, ␪˙ ) is the vector of
centrifugal and Coriolis terms, and G(␪) is the vector of gravity terms.
Complete mathematical equations of motion are presented in the
APPENDIX. These equations, given the initial conditions and the muscle
torques, were integrated forward in time numerically using the RungeKutta algorithm with a constant time step of 0.5 ms to produce the
motion of the segments.
The muscle torques were produced by the following muscle model.
The term “muscle torque” used in this study is equivalent with
“residual torque” used in other studies. The muscles were modeled on
the basis of the model of Stroeve (1996, 1999). This model was the
simplified version of the model of Winters and Stark (1985; see also
Winters 1995). Each muscle contained first-order neuromuscular excitation dynamics (Eq. 2), first-order muscle activation dynamics (e.g.,
calcium kinetics; Eq. 3), a Hill-type force-velocity dependence (Eq.
11), a Gaussian force-length dependence (Eq. 15), a contractile element, a series non-contractile elastic element, and a parallel viscoelastic element. The dynamics of each muscle is described by excitation
(e), activation (a), and length of the contactile element (lce). The
equations of the muscle system, given the initial conditions, were
integrated forward in time numerically using the Runge-Kutta algorithm, given the neural control signal (u). The value of the neural
control signal (u) to each muscle is 0 (inactive) or 1 (active). The P-D
delay was defined as the delay of the neural control signal between the
two muscles (elbow extensor and wrist flexor). The excitation and
activation dynamics were described as follows
ė ⫽ 共u ⫺ e兲/␶ne
ȧ ⫽ 共e ⫺ a兲/␶,
␶⫽
再
␶ac
␶da
(2)
共e ⱖ a兲
共e ⬍ a兲
(3)
where u is the neural input, e, the excitation, a, the activation, ␶ne, the
excitation time constant, ␶ac, the activation time constant, and ␶da, the
de-activation time constant. Figure 2A shows the control signals (u,
thin line of rectangular shape), excitations (e, dashed line), and activations (a, thick line) of the wrist flexor and elbow extensor for a P-D
delay of 70 ms, for example.
FIG. 1. Throwing setup and joint angles. 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). l1 represents the length between the elbow
and the wrist. l2 represents the length between the wrist and
the metacarpophalangeal (MP). The length between the MP
and the fingertip is 0.1 m. See values of the segment parameters in Table 3.
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An experimental study can only examine the actual movements that
are executed by the subjects. On the other hand, a computer simulation study can also examine the arbitrary movements that are produced by artificial motor commands. In this study, we want to clarify
whether the counteractive relationship at the wrist during ball-throwing is determined by the muscle activation patterns of the subjects or
is predominantly determined by the mechanical properties of the
human wrist. Therefore to examine a wider range of motor commands
than those of the subjects obtained in the experiment, we performed
computer simulations.
We employed a two-segment model in the vertical plane [similar to
that of Chowdhary and Challis (1999)] that consisted of the forearm
and hand with only two monoarticular agonist muscles (i.e., elbow
extensor and wrist flexor). Chowdhary and Challis (1999) simulated
ball-throwing by using the forearm-hand model and reported that only
the activities of the agonist muscles produced a movement kinematics
that was almost the same as that of real throwing. It should be noted
that this model was built on some simplifications. The first simplification of this model was that all of the muscles (e.g., flexor carpi
ulnaris, flexor carpi radialis, palmaris longus) that are responsible for
a particular joint rotation (e.g., wrist flexion) were lumped together
into a single idealized muscle (e.g., wrist flexor). This approach is
very common in modeling the musculo-skeletal model when investigators’ concern is not the role of individual muscles but the joint
motion (Winters and Stark 1985). The second simplification was that
there were no antagonist muscles (i.e., elbow flexor and wrist extensor) and finger muscles in the model, therefore some features of the
normal torque profiles in a natural throw will be lost. But, in this
study, this model gave us important knowledge about the relationship
between the mechanical properties of the arm and the dynamical
aspect of the wrist. We examined the “proximal-to-distal delay” (P-D
delay: the delay between onsets of the proximal and distal muscle
contractions) as a major neuromuscular parameter in ball-throwing
movements because the P-D delay has been known to greatly affect
the movement consequence of throwing, i.e., ball velocity or energy
(Alexander 1991; Chowdhary and Challis 1999, 2001; De Lussanet
and Alexander 1997; Herring and Chapman 1992).
The setup of the two-segment model in this study is shown in Fig.
1. The elbow joint was fixed at one point. The horizontal distance
between the target wall and the elbow joint was 2.7 m. The height of
the center of the target was 0.37 m above the height of the elbow joint.
The length of the target was 0.05 m. The initial position was the
balance position where the gravity torque was balanced by the passive
elastic torque and no muscle activity appeared (leftmost figure in Fig.
1). This is the same condition as that instructed to the subjects in our
previous experiment (Hirashima et al. 2003). The positive direction of
the angular displacement was counterclockwise (direction of the arrow in Fig. 1). The positive direction of the ␪1 represents the elbow
flexion. ␪1 was defined relative to the vertical line. The positive
direction of the ␪2 represents the wrist extension. ␪2 was defined
relative to the forearm. Note that ball-throwing is achieved by elbow
extension (negative direction) and wrist flexion (negative direction).
The equation of motion is represented as the second-order ordinal
differential equation, as follows
THE COUNTERACTIVE INTERACTION TORQUE AT THE WRIST IN THROWING
1451
FIG. 2. A: control signal (thin line of rectangular shape), muscle excitation (dashed
line), and muscle activation (thick line) of
the wrist flexor and the elbow extensor for
the proximal-to-distal delay (P-D delay) of
70 ms in the simulation. B: rectified and
averaged electromyograms (EMGs) from
10 trials for the representative subject of the
flexor carpi ulnaris and triceps brachii in our
previous experiment.
l se ⫽ lm ⫺ lce ⫺ lt
(4)
l m ⫽ l r ⫺ rm ␪
(5)
l̇ m ⫽ ⫺rm␪˙
(6)
where lm is the length of the muscle, lce, the length of the contractile
element, lt, the rest length of the series non-contractile element, lr, the
rest length of the muscle, rm, the moment arm, l̇m, the velocity of the
muscle. lse is used to calculate the muscle force (Fce or Fse)
F ce ⫽ Fse共a, lce, lm兲 ⫽
再
0
kse1 共exp关kse2lse兴 ⫺ 1兲
共lse ⱕ 0兲
共lse ⬎ 0兲
(7)
Fmax
k se1 ⫽
exp关SEsh兴 ⫺ 1
(8)
k se2 ⫽ SEsh/SExm
(9)
where SEsh and SExm are the shape and range parameter of the series
element respectively. The muscle forces Fce are used to calculate the
contraction velocities of the contractile element (l̇ce, Eq. 10) from the
inverse force-velocity relationship of the contractile element (Eq. 11).
The contraction velocities of the contractile element (l̇ce) are used to
derive the length of the contractile element (lce). lce is used in Eq. 4 of
the next loop
l̇ ce ⫽
⫺1
vce
F 共a, lce, lm兲 ⫽
冦
再
l̇m
⫺1
Fvce
共a, lce, lm兲
共a ⱕ ␦兲
共a ⬎ ␦兲
Vshvmax共a, lce兲共Fce ⫺ Fiso兲
Fce ⫹ VshFiso
⫺VshlVshvmax共a, lce兲共Fce ⫺ Fiso兲
Fce ⫹ kce1Fiso
vmax共a, lce兲
(10)
共0 ⱕ Fce ⱕ Fiso兲
共Fiso ⬍ Fce ⱕ vmlFiso兲
共vmlFiso ⱕ Fce兲
(11)
where
k ce1 ⫽ ⫺1 ⫺ 共1 ⫹ VshVshl兲共Vml ⫺ 1兲
(12)
v max 共a, lce兲 ⫽ Vvm共1 ⫺ Ver ⫹ VeraFlce共lce兲兲
(13)
F iso ⫽ aFmaxFlce共lce兲
(14)
冋冉
F lce共lce兲 ⫽ exp ⫺
冊册
lce ⫺ lce0
lcesh
where vmax(a, lce) is the maximum velocity of the contractile element,
Fiso is the isometric force at lce, Flce is the relative force of the
contractile element due to the force-length relation, Vsh is the concavity of the Hill curve during shortening, Vshl is the concavity of the
Hill curve during lengthening, lce0 is the optimal length of the contractile element, and lcesh is the width of the Gaussian force-length
curve. The values of all the muscle parameters are shown in Table 1.
Parameters of the muscle model are based on the work of Winters and
Stark (1985) and Stroeve (1996, 1999).
The muscle forces (Fse) are used to calculate the muscle torques
(T1, elbow muscle torque; T2, wrist muscle torque). In this study,
moment arms are constant
T 1 ⫽ Fse1rm1 ⫹ Tp1
(16)
T 2 ⫽ Fse2rm2 ⫹ Tp2
(17)
where rm1 is the moment arm of the elbow extensor on the elbow
joint, rm2 is the moment arm of the wrist flexor on the wrist joint, and
Tp1 and Tp2 are the passive viscoelastic torques at the elbow and wrist,
respectively. It should be noted that the muscle torques include not
only the active muscle torque but also the passive torque from a
parallel viscoelastic element. The forward simulation with this model
and the inverse dynamics analysis complementarily gave important
TABLE
1. Parameters of the muscle model
Parameter
Elbow Extensor
Wrist Flexor
Unit
␶ne
␶ac
␶da
␦
rm1
rm2
Fmax
Tmax
lr
lt
lcesh
lce0
Ver
Vml
Vvm
Vsh
Vshl
SExm
SEsh
0.04
0.01
0.05
10⫺4
⫺0.03
0
1500
45
0.15
0.25lr
0.25lr
0.75lr
0.5
1.3
6lce0
0.3
0.5
0.05lr
3.0
0.04
0.01
0.05
10⫺4
0
⫺0.03
400
12
0.15
0.25lr
0.25lr
0.75lr
0.5
1.3
6lce0
0.3
0.5
0.05lr
3.0
s
s
s
—
m
m
N
N䡠m
m
m
m
m
—
—
m/s
—
—
m
—
2
J Neurophysiol • VOL
(15)
Parameters of the muscle model are based on the work of Winters and Stark
(1985) and Stroeve (1996, 1999).
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The numerical calculation of the forward simulation starts by
calculating the stretch length of the series non-contractile element (lse)
1452
HIRASHIMA, OHGANE, KUDO, HASE, AND OHTSUKI
insights in understanding the motor-control strategy. By using inverse
dynamics analysis alone, the active muscle torque cannot be differentiated from the passive viscoelastic torque. By using this model, the
active muscle torque and the passive viscoelastic torque can be dealt
with separately. The passive torque is described as the combination of
elastic and viscous component as follows
T pj ⫽ ⫺Kj共␪j ⫺ ␪rj兲 ⫺ sgn 共␪j ⫺ ␪rj兲
Tmaxj
exp关PEshj兴
冉 冋 冏 冏册 冊
⫻ exp
PEshj
PExmj
␪j ⫺ ␪rj
⫺ 1 共elastic兲
⫺B j ␪˙ j
(viscous)
(18)
Simulation 1 (ball-throwing by the normal forearm-hand
model)
The purpose of simulation 1 was to examine whether neuromuscular activity can result in the assistive relationship between the interaction torque and muscle torque at the normal human wrist. At time 0,
the neural input signal to the elbow extensor took a value of 1 (active).
After a certain delay (i.e., P-D delay), the neural input signal to the
wrist flexor took a value of 1 (active). The active duration was 250 ms
for both muscles (see Fig. 2A).
The ball was released in the following way. The ball release angle
is defined as the angle that is rotated 20° above from the velocity
vector of the metacarpophalangeal (MP) joint. The degree of 20 was
confirmed in a previous experiment. The magnitude of the velocity
vector of the ball at the ball-release time is defined to be the same as
the magnitude of the velocity vector of the fingertip. If a ball released
at a certain time hit the target, that time was defined as the optimal
ball-release time. Often there were two or three optimal ball-release
times. In such cases, the ball-release time by which the largest
horizontal ball velocity was obtained was defined as the optimal
ball-release time. In this way, one throwing movement was simulated.
This procedure was performed for each P-D delay that was system-
FIG. 3. The passive elastic torque at the wrist of four types of rest angle
plotted against the wrist angle (␪2). Rest angle of normal model is referred to
as 0°. For example, consider the case of the wrist angle (␪2) is 60° (see *); the
passive elastic torque does not exist in the virtual model with the rest angle of
60°, while the flexion passive elastic torque (⫺2.65 Nm) is produced in the
normal model with the rest angle of 0°.
atically varied from 0 to 200 ms (i.e., 0, 10, 20, 30 . . . 190, and 200
ms). In total, 21 throwing movements were simulated in simulation 1.
Simulation 2 (ball-throwing by the virtual forearm-hand
model)
By considering the results of simulation 1, we could line up two
candidates as the determinant of the counteractive relationship. They
were the rest angle (rest angle is the angle where the passive elastic
torque is 0) of the wrist and the length and mass of the hand. The
purpose of simulation 2 was to manipulate these two parameters and
simulate the ball-throwing movements of the virtual forearm-hand
model. We prepared four types of length and mass of the hand (75,
100, 125, and 150% of the normal hand) to change the moment of
inertia of the hand. Because the procedure and setup of simulation 2
were the same as those of simulation 1 except for the model used, we
did not adopt extreme parameters such as 50 or 200% due to the size
mismatch between the setup (Fig. 1) and the virtual model with
extreme parameters. We also prepared four types of rest angle of the
wrist (0, 30, 60, and 90°). Changing the rest angle means changing the
profile of the passive elastic torque at the wrist. Figure 3 shows the
passive elastic torque at the wrist of four types of rest angle plotted
against the wrist angle (␪2). For example, consider a case in which the
wrist angle (␪2) is 60° (see * in Fig. 3); the passive elastic torque does
not exist in the virtual model with a rest angle of 60°, while the flexion
passive elastic torque (2.65 N 䡠 m) is produced in the normal model
with a rest angle of 0°. Overall, we used 15 (4 ⫻ 4 ⫺ 1) virtual
forearm-hand models, as one model (100% and 0°) is normal. For
each virtual model, we simulated 21 throwing movements by systemTABLE
TABLE
2. Parameters of the passive viscoelastic elements
Parameter
Elbow
Wrist
Unit
K
B
␪r
PExm
PEsh
1.5
0.2
80 ⫹ 90
␲/2
9.0
1.0
0.1
0
␲/2
6.0
Nm/rad
Nms/rad
degree
rad
—
Parameters of the passive viscoelastic elements are based on the work of
Winters and Stark (1985) and Stroeve (1996, 1999).
J Neurophysiol • VOL
3. Parameters of the segment
Parameter
Forearm
Hand
Unit
li
mi
Ii
ri
0.26
1.15
m1(0.279 ⫻ l1)2
0.415 ⫻ l1
0.085
0.432 ⫹ 0.136*
m2(0.519 ⫻ l2)2
0.891 ⫻ l2
m
kg
kgm2
m
li and mi are based on one subject of our previous experiment (Hirashima et
al. 2003). The anthropometric data are based on data on Japanese obtained by
Ae et al. (1992) and reported in a paper by Yokoi et al. (1998).
* The mass of the ball (0.136 kg) is included in the hand mass.
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where Kj is the joint stiffness, Bj is the joint damping, sgn ⵹ is the
signum function (which returns ⫹1 when the content ⬎ 0; which
returns ⫺1 when the content ⬍0; which returns 0 when the content ⫽
0), PEshj and PExmj are the shape and range parameters of the passive
torque, respectively, Tmaxj is the maximum torque of both joints, and
␪rj is the rest angle (where the passive elastic torque is 0, j ⫽ 1: elbow,
2: wrist). Note that the rest angle is different from the angle of the
initial position (leftmost figure in Fig. 1). Values of parameters of the
passive torque are based on the work of Winters and Stark (1985) and
Stroeve (1996, 1999) (see Table 2). The ranges of motion at the joints
were limited by these passive elasticities (see Fig. 3). This forearmhand model can be decelerated by these passive elasticities, although
this model does not contain the antagonist muscles. The muscle
torques (T1 and T2) were used as input to the equations of motion of
each segment to obtain the angular accelerations (Eq. 1). The anthropometric data used in the model are based on data on Japanese
obtained by Ae et al. (1992) and reported in a paper by Yokoi et al.
(1998). These data are shown in Table 3.
THE COUNTERACTIVE INTERACTION TORQUE AT THE WRIST IN THROWING
atically changing the P-D delay from 0 to 200 ms. In total, 315 (15 ⫻
21) throwing movements were simulated in simulation 2.
where T is the optimal ball-release time, I(t) and M(t) was calculated
as follows
Inverse dynamics
I共t兲 ⫽
Elbow
NETe ⫽ ⫹␪¨ 1[I1 ⫹ I2 ⫹ m2l21 ⫹ m1r21 ⫹ m2r22 ⫹ 2m2l1r2 cos ␪2]
INTe ⫽ ⫺␪¨ 2[I2 ⫹ m2r22 ⫹ m2l1r2 cos ␪2] (inertial torquee)
⫹␪˙ 22[m2l1r2 sin ␪2] (centripetal torquee)
⫹␪˙ 1␪˙ 2[2m2l1r2 sin ␪2] (Coriolis torquee)
GRAe ⫽ ⫺g[(m2l1 ⫹ m1r1) sin ␪1 ⫹ m2r2 sin (␪1 ⫹ ␪2)]
MUSe ⫽ NETe ⫺ INTe ⫺ GRAe
Wrist
NETw ⫽ ⫹␪¨ 2[I2 ⫹ m2r22]
INTw ⫽ ⫺␪¨ 1[I2 ⫹ m2r22 ⫹ m2l1r2 cos ␪2] (inertial torquew)
⫺␪˙ 21[m2l1r2 sin ␪2] (centripetal torquew)
GRAw ⫽ ⫺g[m2r2 sin (␪1 ⫹ ␪2)]
MUSw ⫽ NETw ⫺ INTw ⫺ GRAw
where Ii is moment of inertia about the center of gravity, ri is distance
to center of mass from its proximal joint, li is length, mi is mass (i ⫽
1: forearm, 2: hand). The mass of the ball is included in the hand mass.
Index of coordination between the interaction torque
and the muscle torque
There are two representative coordination relationships between the
interaction torque and the muscle torque: “counteractive” and “assistive” relationship (for more details, see Hirashima et al. 2003). 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 throwing movement was calculated as follows
IOCIM ⫽
冕
冕
T
I共t兲dt
0
T
M共t兲dt
0
J Neurophysiol • VOL
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 0 to T as a whole. The magnitude of the IOCIM 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.
Electromyography
We recorded real muscle activities during the two-joint throwing
when throwing fast in our previous experiment (for more details, see
Hirashima et al. 2003).
Electromyographic (EMG) activity was recorded from five subjects
(mean age: 23.2 yr). 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.
EMG activities of the triceps brachii (TB; elbow extensor) and
flexor carpi ulnaris (FCU; wrist flexor) were recorded by bipolar
Ag-AgCl disc surface electrodes (8 mm diam) attached 2 cm apart
along the longitudinal axis of the muscle belly. The surface of the skin
was treated with alcohol and rubbed with fine sandpaper to reduce
inter-electrode resistance before the electrodes were attached. EMG
signals were digitized at 1,000 Hz for 3,000 ms and telemetered
(SYNA ACT, NEC) to an analog-digital converter (NR-2000, Keyence); they were then recorded in a personal computer (ThinkPad-240,
IBM). To identify the ball-release time, motions of the hand and ball
were recorded at 200 Hz using a high-speed video camera (HAS200R, Ditect). Synchronization of the video pictures and the EMG
signals was accomplished by means of an electronic pulse that simultaneously marked the margin of the picture and the EMG signals.
EMG signals were high-pass filtered (cut-off frequency ⫽ 20 Hz),
full-wave-rectified, and finally low-pass filtered (cut-off frequency ⫽
50 Hz). The onset of the EMG was defined as the time point at which
EMG amplitude exceeded a threshold and after which EMG amplitude remained above that level continuously for ⱖ25 ms (Hodges and
Bui 1996). The threshold value was set to the mean background
activity plus 6 SD. The background activity was averaged between at
400 ms prior to the ball-release time and at 300 ms prior to the
ball-release time.
RESULTS
Simulation 1
In simulation 1, we simulated the throwing motions by using
a normal human forearm-hand model. Figure 4 shows the
simulation results of P-D delays of 0, 60, and 120 ms, which
are stick pictures of the simulated throwing (Fig. 4A), joint
angle (Fig. 4B), angular velocity (Fig. 4C), four torques at the
wrist (i.e., NETw, INTw, GRAw, and MUSw; Fig. 4D), details
of the INTw (i.e., INTw ⫽ inertial torquew ⫹ centripetal
torquew; Fig. 4E), and details of the MUSw (i.e., MUSw ⫽
active torquew ⫹ viscous torquew ⫹ elastic torquew; Fig. 4F).
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By these forward simulations, we obtained the time-series data of
the joint angles, angular velocities, and angular accelerations at the
elbow and wrist joints for each simulation. By using these data, we
calculated the time series of the net torque (NET), gravity torque
(GRA), interaction torque (INT), and muscle torque (MUS) at the
wrist joint. 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). 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 active
muscle contraction acting at the joint but also the passive contributions by muscles, tendons, ligaments, articular capsules, and other
connective tissues. NET, GRA, INT, and MUS at the elbow and wrist
are described as follows
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P-D DELAY OF 0 MS. When the P-D delay was 0 ms (Fig. 4,
left), at first, elbow extension and wrist flexion occurred (Fig.
4C). The interaction torque at the wrist was in the opposite
direction (extension) throughout the motion, and the flexion net
torque was small (Fig. 4D). This caused small wrist angular
velocity and small horizontal ball velocity at the ball-release
time (5.4 m/s). This counteractive interaction torque at the
wrist was mainly due to the inertial torque (Fig. 4E). The
centripetal torque at the wrist was very small until the
ball-release time (Fig. 4E) because the wrist angle (␪2) was
⬃0° (i.e., sin ␪2 was almost 0; Fig. 4B) and the centripetal
torque at the wrist contains sin ␪2 (centripetal torquew ⫽
⫺␪˙ 21[m2l1r2 sin ␪2]).
P-D DELAY OF 60 MS. When the P-D delay was 60 ms (Fig. 4,
middle), at first, elbow extension occurred (Fig. 4C). This
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THE COUNTERACTIVE INTERACTION TORQUE AT THE WRIST IN THROWING
ENTIRE SIMULATION 1 WITH A NORMAL FOREARM-HAND MODEL.
Figure 4G shows the relationship between the interaction
torque and muscle torque of P-D delays of 0, 20, 40, 60, 80,
100, 120, and 140 ms. The IOCIM is also shown in the
right-bottom corner of each panel. The IOCIM is plotted
against the P-D delay in Fig. 4H. When the optimal ball-release
time occurred before the onset of the wrist flexor, the plot in
Fig. 4, H and I, was removed. The IOCIM never got positive
sign, although the longer the P-D delay became, the smaller the
counteractive level of the IOCIM became. The assistive relationship would be prevented by the fact that the flexion centripetal torque at the wrist (centripetal torquew ⫽ ⫺␪˙ 21[m2l1r2
sin ␪2]) is very small, which is caused by the small wrist joint
angle (␪2) and the small value of m2l1r2.
REAL MUSCLE ACTIVITIES. As for real muscle activities, first,
the TB started its activation [across subjects, 172.4 ⫾ 28.1
(SD) ms before the ball-release time], and then, the FCU
started its activation (across subjects, 97.1 ⫾ 6.6 ms before the
ball-release time; Fig. 2B). The paired t-test performed on the
EMG onset indicated significant differences between the TB
and FCU (P ⬍ 0.005). The mean difference of the onset
between TB and FCU was 75.3 ⫾ 27.2 ms. Figure 4I shows the
horizontal ball velocity plotted against the P-D delay of the
simulation. The P-D delay adopted by our subjects was near
the P-D delay (70 ms) by which the largest horizontal ball
velocity was obtained in the simulation.
Simulation 2
We hypothesized that a larger rest angle of the wrist and a
larger value of m2l1r2 can produce a larger flexion centripetal
torque at the wrist and finally produce the assistive interaction
torque. We manipulated the rest angle of the wrist, the mass of
the hand (m2) and the length of the hand (l2; r2 ⫽ 0.891 ⫻ l2).
REST ANGLE, 60°; LENGTH AND MASS, 100%. First, we show the
simulation results of a rest angle of 60° and length and mass of
100%, changing only the rest angle. Figure 5 shows simulation
data of P-D delays of 0, 60, and 120 ms. As we expected, the
flexion centripetal torque (Fig. 5E) was larger than that of the
normal human forearm-hand model (Fig. 4E) in any P-D delay.
The longer the P-D delay became, the larger the flexion centripetal torque became. This was because the wrist angle (␪2)
was ⬃90° (i.e., sin ␪2 was almost 1), especially in a P-D delay
of 120 ms (Fig. 5B). In addition, the longer the P-D delay
became, the smaller the extension inertial torque became (Fig.
5E). The large rest angle affected the interaction torque and
enabled the CNS to produce the flexion interaction torque at
the wrist (Fig. 5D).
Figure 5G shows the relationship between the interaction
torque and muscle torque of P-D delays of 0, 20, 40, 60, 80,
100, 120, and 140 ms in this virtual model. The IOCIM is also
shown in the right-bottom corner of each panel. The longer the
P-D delay became, the smaller the counteractive level of
IOCIM became up to a P-D delay of 60 ms. Although the
flexion interaction torque was produced for P-D delays of 80,
100, 120, and 140 ms, the IOCIM never got a positive sign
(Fig. 5H). This was because the direction of the wrist muscle
torque resulted in extension, which was caused by the extension viscous torque (Fig. 5F, right). Even if the flexion interaction torque produced the flexion net torque and wrist flexion
angular velocity, the passive viscous torque counteracted it.
The viscosity at the wrist joint produced comparable torque
with the active wrist muscle torque (Fig. 5F, middle and right).
As the assistive duration during which both the interaction and
muscle torques were in the flexion was short (from ⬃75 ms to
FIG. 4. The simulation data of the P-D delay of 0 ms (left), 60 ms (middle), and 120 ms (right) with using the normal human
forearm-hand model. A: the stick picture with bold line is the initial configuration of the arm model. The other stick pictures are
configurations at ⫺100, ⫺75, ⫺50, ⫺25, and 0 ms prior to the ball-release time. Paths of the metacarpophalangeal (MP) joint and
the wrist joint and the ball thrown at the optimal ball-release time are drawn. B: joint angle. C: angular velocity. D: the net torque
(thick line), gravity torque (dashed line), muscle torque (thin line), and interaction torque (dotted line) at the wrist. Open arrow
indicates the onset of wrist flexor. E: the inertial torque (dashed line) and centripetal torque (solid line) at the wrist. F: the active
torque (thin line), passive viscous torque (thick line), and passive elastic torque (dashed line) at the wrist. The vertical line of each
panel represent the ball-release time. Four vertical dotted lines represent the time point of ⫺100, ⫺75, ⫺50, and ⫺25 ms prior to
the ball-release time. G: the relationship between the wrist interaction torque and wrist muscle torque of the P-D delay of 0, 20,
40, 60, 80, 100, 120, and 140 ms. The index of coordination between the interaction torque and the muscle torque (IOCIM; see
METHODS) was also shown in the right-bottom corner of each panel. H: the IOCIM is plotted against the P-D delay. I: the horizontal
velocity of the ball thrown at the optimal ball-release time is plotted against the P-D delay. When the optimal ball-release time
occurred before the onset of the wrist flexor, the plot in 4H and I was removed.
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caused wrist extension interaction torque from 0 to 75 ms (Fig.
4D). This wrist extension interaction torque overcame the wrist
flexion muscle torque and produced the wrist extension. The
wrist flexion muscle torque in this phase was due to the viscous
and elastic torques (Fig. 4, D and F). The horizontal ball
velocity at the ball-release time became larger (6.7 m/s) for the
following two reasons. 1) As the wrist counteractive interaction torque was small, the muscle torque could produce a larger
net torque, especially from 60 to 100 ms (Fig. 4D). 2) As the
wrist flexion was delayed from the elbow extension, the ballrelease time was also delayed. The acceleration time for both
the wrist and elbow was prolonged, and, as a consequence,
larger angular velocity for both joints occurred. However,
assistive interaction torque did not occur (Fig. 4D), although
the counteractive interaction torque at the wrist was smaller
than that of the P-D delay of 0 ms. This was because the
centripetal torque at the wrist was very small until the ballrelease time as well as for the P-D delay of 0 ms (Fig. 4E).
P-D DELAY OF 120 MS. When the P-D delay was 120 ms (Fig.
4, right), at first, elbow extension and wrist extension occurred
(Fig. 4C). The P-D delay was so long that the active muscle
torque’s contribution to the wrist flexion muscle torque was
little (Fig. 4F). Most of the wrist flexion muscle torque came
from the passive elastic torque (Fig. 4F). As a consequence, the
wrist flexion muscle torque was smaller than that for a P-D
delay of 60 ms (Fig. 4D). However, the performance was not
bad (horizontal ball velocity ⫽ 6.2 m/s) because the counteractive interaction torque at the wrist became smaller (Fig. 4D).
This was caused by the fact that the extension inertial torque
became smaller and the flexion centripetal torque at the wrist
became slightly larger (Fig. 4E).
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FIG. 5. The simulation data of the P-D delay of 0 ms (left), 60 ms (middle), and 120 ms (right) with the rest angle of 60° and
100% of length and mass of the hand. A: the stick picture with bold line is the initial configuration of the arm model. The other
stick pictures are configurations at ⫺100, ⫺75, ⫺50, ⫺25, 0 ms prior to the ball-release time. Paths of the MP joint and the wrist
joint and the ball thrown at the optimal ball-release time are drawn. B: joint angle. C: angular velocity. D: the net torque (thick line),
gravity torque (dashed line), muscle torque (thin line), and interaction torque (dotted line) at the wrist. Open arrow indicates the
onset of wrist flexor. E: the inertial torque (dashed line) and centripetal torque (solid line) at the wrist. F: the active torque (thin
line), passive viscous torque (thick line), and passive elastic torque (dashed line) at the wrist. The vertical line of each panel
represent the ball-release time. Four vertical dotted lines represent the time point of ⫺100, ⫺75, ⫺50, and ⫺25 ms prior to the
ball-release time. G: the relationship between the wrist interaction torque and wrist muscle torque of the P-D delay of 0, 20, 40,
60, 80, 100, 120, and 140 ms. The IOCIM (see METHODS) was also shown in the right-bottom corner of each panel. H: the IOCIM
is plotted against the P-D delay. When the optimal ball-release time occurred before the onset of the wrist flexor, the plot in H was
removed.
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THE COUNTERACTIVE INTERACTION TORQUE AT THE WRIST IN THROWING
⬃150 ms, P-D delays of 80, 100, 120, and 140 ms in Fig. 5G),
the IOCIM indicating the coordination relationship from 0 ms
to the ball-release time as a whole never got a positive sign in
any P-D delay.
REST ANGLE, 60°; LENGTH AND MASS, 150%. Third, we show the
simulation data of a rest angle of 60° and length and mass of
150%, changing both parameters. The longer the P-D delay
became, the smaller the extension inertial torque at the wrist
became and the larger the flexion centripetal torque at the wrist
became (Fig. 7E). As a result, large flexion interaction torque
at the wrist occurred (Fig. 7D, right). By virtue of the large
length and mass of the hand, the flexion interaction torque at
the wrist in this model was larger than that in the model of 60°
and 100% (compare Fig. 7D with 5D). In addition, the ballrelease time was more delayed than that in the model of 60°
and 100% due to the large moment of inertia of the hand
(compare Fig. 7G with 5G), and the assistive duration was also
prolonged. As the flexion interaction torque at the wrist
strongly assisted the flexion muscle torque for a long time
(from ⬃100 to ⬃200 ms, P-D delays of 80, 100, 120, and 140
ms in Fig. 7G), the IOCIM got a positive sign for some P-D
delays (Fig. 7H).
ENTIRE SIMULATION 2 WITH VIRTUAL FOREARM-HAND MODELS.
Figure 8 shows the IOCIM plotted against the P-D delay of all
16 forearm-hand models. In line with the results of the three
cases described in the preceding text, the longer the P-D delay
became, the smaller the counteractive level became. In the five
models (surrounded by the thick line) of the 16 models in Fig.
8, the IOCIM showed a positive sign for some P-D delays. In
general, both the large rest angle and large moment of inertia
(length and mass) of the hand contributed to a large value of
IOCIM. The larger rest angle than normal could not realize the
assistive relationship only by itself (e.g., Fig. 5H). The larger
moment of inertia than normal could not realize the assistive
relationship only by itself either (e.g., Fig. 6H). The positive
value of IOCIM at the wrist in two-joint throwing requires that
the following two conditions be simultaneously achieved: 1)
large flexion centripetal torque that is mainly caused by the
large rest angle and 2) long time of assistive duration that is
caused by the large moment of inertia (large length and mass)
of the hand. This suggested that only the nonphysiological
range of two parameters can produce the assistive relationship
between the interaction torque and muscle torque in two-joint
throwing.
J Neurophysiol • VOL
DISCUSSION
In this study, we focused on the apparently unexpected
phenomenon that appeared in our previous experiment
(Hirashima et al. 2003). The phenomenon was that the interaction torque at the wrist joint was always counteractive to the
wrist muscle torque irrespective of the ball speed. It is unclear
whether the counteractive relationship at the wrist during ballthrowing is caused by the neural contribution or is predominantly determined by the musculoskeletal mechanical properties of the human wrist. We hypothesized that the counteractive
relationship at the wrist has a predestined nature that is constrained by the mechanical properties of the human arm because of the following: if the CNS could adopt an assistive
relationship irrespectively of the mechanical property, the CNS
would have to use the assistive relationship when throwing
fast, as the assistive relationship is advantageous for generating
large angular velocity at the joint. In this study, we addressed
this issue by using a computer simulation of two-joint throwing
and found that the counteractive relationship tended to occur at
the wrist joint. Here, we discuss the three determinants for the
counteractive relationship at the wrist joint in ball-throwing
movements. They are the wrist rest angle, the length and mass
of a normal hand, and viscoelasticity at the wrist.
Wrist rest angle
Although we used the simplified model with only two monoarticular agonist muscles and manipulated only the P-D delay
as a neuromuscular activity, it was enough to suggest that it is
very difficult for the CNS to produce the flexion interaction
torque at the wrist because this torque is mainly prevented by
a normal wrist rest angle (0°) that can not be changed by the
CNS. Although the CNS can reduce the extension inertial
torque by increasing the P-D delay, it cannot produce a large
flexion centripetal torque in a normal forearm-hand model. The
normal wrist rest angle of 0°, of course, exists in any movement; therefore, the normal wrist rest angle is likely to function
as a strong determinant for the counteractive relationship in
other swing motions. Actually, the wrist interaction torque
during three-joint throwing was always counteractive to the
muscle torque (Hirashima et al. 2003).
However, it would be risky to conclude that there is always
a counteractive relationship in any swing motion because, in
multi-joint movements including more joints (e.g., Cooper et
al. 2000), the equation of the wrist interaction torque has many
more terms, and the behavior of such a complicated nonlinear
system cannot be predicted. To confirm the existence of the
persistent counteractive relationship at the wrist in any swing
motion, it is necessary to examine more complicated multijoint swing motions, such as baseball pitching with the whole
body (Feltner 1989; Fujii and Hubbard 2002; Hirashima et al.
2002; Matsuo et al. 2002), in greater depth.
Length and mass of the hand
Simulation 2 indicated that large length and mass of the hand
could not realize the assistive relationship by itself (Fig. 8,
bottom). However, the larger the length and mass of the hand
became, the smaller the counteractive level became, which is
due to elongation of the assistive duration (compare Fig. 6G
with 4G). Therefore there remains the possibility that much
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Second, we show the
simulation data of a rest angle of 0° and length and mass of
150%, changing only the length and mass of the hand (Fig. 6).
The longer the P-D delay became, the smaller the counteractive
level became (Fig. 6G). The wrist flexion centripetal torque
was produced in a later phase of throwing for a P-D delay of
120 ms due to the large value of m2l1r2 (Fig. 6E, right).
However, this large value of m2l1r2 also produced a large wrist
extension inertial torque in the initial phase. Therefore the
interaction torque from 0 ms to the ball-release time as a whole
was counteractive, and the IOCIM never got a positive sign in
any P-D delay (Fig. 6, G and H). It should also be noted that
the ball-release time was more delayed than that of the normal
model due to the large moment of inertia (large length and
mass) of the hand (compare Fig. 6G with 4G).
REST ANGLE, 0°; LENGTH AND MASS, 150%.
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FIG. 6. The simulation data of the P-D delay of 0 ms (left), 60 ms (middle), and 120 ms (right) with the rest angle of 0° and 150%
of length and mass of the hand. A: the stick picture with bold line is the initial configuration of the arm model. The other stick pictures
are configurations at ⫺100, ⫺75, ⫺50, ⫺25, and 0 ms prior to the ball-release time. Paths of the MP joint and the wrist joint and the
ball thrown at the optimal ball-release time are drawn. B: joint angle. C: angular velocity. D: net torque (thick line), gravity torque (dashed
line), muscle torque (thin line), and interaction torque (dotted line) at the wrist. Open arrow indicates the onset of wrist flexor. E: inertial
torque (dashed line) and centripetal torque (solid line) at the wrist. F: active torque (thin line), passive viscous torque (thick line), and
passive elastic torque (dashed line) at the wrist. The vertical line of each panel represent the ball-release time. Four vertical dotted lines
represent the time point of ⫺100, ⫺75, ⫺50, and ⫺25 ms prior to the ball-release time. G: relationship between the wrist interaction
torque and wrist muscle torque of the P-D delay of 0, 20, 40, 60, 80, 100, 120, and 140 ms. The IOCIM (see METHODS) was also shown
in the right-bottom corner of each panel. H: the IOCIM is plotted against the P-D delay.
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FIG. 7. The simulation data of the P-D delay of 0 ms (left), 60 ms (middle), and 120 ms (right) with the rest angle of 60° and
150% of length and mass of the hand. A: the stick picture with bold line is the initial configuration of the arm model. The other
stick pictures are configurations at ⫺100, ⫺75, ⫺50, ⫺25, and 0 ms prior to the ball-release time. Paths of the MP joint and the
wrist joint and the ball thrown at the optimal ball-release time are drawn. B: joint angle. C: angular velocity. D: net torque (thick
line), gravity torque (dashed line), muscle torque (thin line), and interaction torque (dotted line) at the wrist. Open arrow indicates
the onset of wrist flexor. E: inertial torque (dashed line) and centripetal torque (solid line) at the wrist. F: active torque (thin line),
passive viscous torque (thick line), and passive elastic torque (dashed line) at the wrist. The vertical line of each panel represent
the ball-release time. Four vertical dotted lines represent the time point of ⫺100, ⫺75, ⫺50, and ⫺25 ms prior to the ball-release
time. G: relationship between the wrist interaction torque and wrist muscle torque of the P-D delay of 0, 20, 40, 60, 80, 100, 120,
and 140 ms. The IOCIM (see METHODS) was also shown in the right-bottom corner of each panel. H: the IOCIM is plotted against
the P-D delay.
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Role of the wrist
larger length and mass of the hand produced the assistive
relationship by itself. If so, humans can change the dynamical
aspect of the wrist by, for example, grasping a tennis racket
that is much longer than 150%. Therefore comparing the wrist
motion of ball-throwing with that of the tennis stroke may give
important insights into motor adaptation for different dynamical aspects of the wrist.
Viscoelasticity at the wrist
The viscoelastic torque at the wrist also plays an important
role for maintaining the counteractive relationship. The viscosity at the wrist produces a considerable torque that is comparable to the active wrist muscle torque or the wrist interaction
torque. Even if the wrist flexion interaction torque was produced first and then the wrist flexion net torque was produced,
the extension viscous torque would counteract the wrist flexion
interaction torque (i.e., a counteractive relationship, see Fig. 5,
D and F, right). The results of this simulation study gave some
insight into the experimental data, suggesting, for example,
that the viscous torque at the wrist considerably contributed to
the wrist extension muscle torque during ⬃25 ms just before
the ball-release time in the slow and medium conditions in our
previous experiments (see Figs. 2D and 4D in Hirashima et al.
2003) because the wrist angular velocity during this phase had
been large (⫺500 ⬃ ⫺800°/s). It would be wrong to conclude
that the wrist extension muscle torque during this phase was
produced by the active contribution from the wrist extensors.
The elastic torque also contributes to forming the counteractive relationship. The fact that our subjects adopted a P-D
delay of ⬃75 ms indicated that the wrist flexion muscle torque
before the onset of the wrist flexor (from about ⫺200 to ⫺100
ms in Fig. 2D of Hirashima et al. 2003) was mainly produced
not by the contraction of the active wrist flexor but by the
passive elastic torque (see also Fig. 4, D and F, middle and
right). The wrist flexion muscle torque in this initial phase
counteracted the extension interaction torque.
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FIG. 8. The IOCIM plotted against the P-D delay of all 16 forearm-hand
models. In 5 (surrounded by the thick line) of the 16 models, the IOCIM
showed positive sign for some P-D delays.
Here, we discuss the role of the wrist joint considering the
disadvantage and advantage of the counteractive relationship at
the wrist. As the length of the hand is shorter than that of the
forearm plus hand, the wrist angular velocity is less effective
for increasing ball speed than the angular velocity of the elbow.
In addition, our present results indicated that it is difficult for
the CNS to produce an assistive relationship at the wrist that
helps to generate a large wrist angular velocity. Therefore the
wrist is not adequate to generate large ball velocity in ballthrowing movements either kinematically or dynamically.
Considering this from a different point of view, the counteractive relationship between the interaction torque and muscle torque is advantageous for keeping the state (angle and
angular velocity) of the wrist joint stable. The wrist net torque
does not exceed the limit of the wrist muscle torque itself by
virtue of the counteractive interaction torque. The first advantage of this is to avoid wrist injury due to excessive extension
or flexion. The second advantage would be that the forceproducing capacity of the extrinsic finger muscles does not
change at high speed. As the extrinsic finger muscles cross the
wrist joint, the wrist motion 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). If the muscle torque and interaction torque cooperatively produced much larger angular velocity at the wrist, the
force-producing capacity of the extrinsic finger muscles would
change at higher speed, and the finger grip control would be
more difficult.
In summary, the wrist is equipped with a self-defense mechanism and provides a relatively stable base for extrinsic finger
control, which would be very advantageous when accurate
finger grip control is required during multi-joint upper limb
movements such as ball-throwing (Hore et al. 1996a,b, 1999,
2001; Timmann et al. 1999 –2001) and transporting or lifting
an object (Flanagan et al. 1993; Flanagan and Tresilian 1994;
Flanagan and Wing 1993, 1995, 1997; Johansson and Westling
1984; Kinoshita et al. 1997; Nowak et al. 2001, 2002).
During the reaching movements in which wrist motion is not
necessarily required, in particular, humans keep the wrist joint
almost motionless (Cruse et al. 1993; Dean and Bruwer 1994;
Koshland and Hasan 1994; Koshland et al. 2000), which would
lead to the simplification of extrinsic finger control because the
force-producing capacity of the extrinsic finger muscles does
not change throughout the movement. Koshland et al. (2000)
reported that the role of wrist muscles during a reaching task
may be to appropriately counteract the interaction torque that
was produced by the motions of the proximal joints. Koshland
et al. (2000) also reported that no case of reaching has been
described in which wrist muscle torques assisted interaction
torques, giving several examples (Dounskaia et al. 1998; Ghez
et al. 1996; Virji-Babul and Cooke 1995). This counteractive
relationship during reaching movements may be accomplished
by the CNS (Koshland et al. 2000), probably for the purpose of
simplifying extrinsic finger grip control or keeping a comfortable wrist posture (Rossetti et al. 1994).
During the ball-throwing movements in which the wrist
motion can contribute to the movement performance (i.e., ball
speed), humans keep the time series of the wrist kinematics
relatively constant irrespectively of the ball speed (Hirashima
THE COUNTERACTIVE INTERACTION TORQUE AT THE WRIST IN THROWING
et al. 2003). The present simulation study indicates that this
wrist kinematics was also accomplished by a counteractive
relationship that was destined to occur due to the mechanical
properties of the human arm.
CNS and the mechanical property
APPENDIX
Equations of motion of the normal forearm-hand model
M共␪兲␪¨ ⫽ 关T ⫺ V共␪, ␪˙ 兲 ⫺ G共␪兲兴
M共␪兲 ⫽
␪¨ ⫽
T⫽
冉
冊
冉 冊
冉
冊
M11
M21
M12
M22
␪¨ 1
␪¨ 2
T1 ⫺ T2
T2
J Neurophysiol • VOL
G共␪兲 ⫽
冉 冊
冉 冊
V1
V2
G1
G2
where ␪ is the vector of the joint angle, ␪˙ is the vector of joint angular
velocity, ␪¨ is the vector of joint angular acceleration, M(␪) is the
inertia matrix, V(␪, ␪˙ ) is the vector of centrifugal and Coriolis terms,
G(␪) is the vector of gravity terms, T1 is the elbow muscle torque (see
Eq. 16), and T2 is the wrist muscle torque (see Eq. 17). They are
described as follows
M11
M12
V1
G1
M21
M22
V2
G2
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
I1 ⫹ l21m2 ⫹ m1r21 ⫹ l1m2r2 cos ␪2
l1m2r2 cos ␪2
⫺␪˙ 1␪˙ 2(2l1m2r2 sin ␪2) ⫺ ␪˙ 21(l1m2r2 sin ␪2) ⫺ ␪˙ 22(l1m2r2 sin ␪2)
g(l1m2 sin ␪1 ⫹ m1r1 sin ␪1)
I2 ⫹ m2r22 ⫹ l1m2r2 cos ␪2
I2 ⫹ m2r22
␪˙ 21(l1m2r2 sin ␪2)
g(m2r2 sin (␪1 ⫹ ␪2))
where mi is the mass, Ii is the inertia, li is the length, ri is the distance
to the center of mass from its proximal joint, of both segments (i ⫽ 1:
forearm, 2: hand). Values of all parameters are shown in Table 3.
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