ELSEVIER
Human
Movement
Science
13 (1994) 611-634
Control properties induced by the existence of antagonistic
pairs of bi-articular muscles - Mechanical engineering
model analyses
Minayori Kumamoto
*, Toru Oshima, Tomohisa Yamamoto
Laboratory of Human Motor Control, Faculty of Engineering, Toyama Prefectural University, Kosugi,
Toyama 939-03, Japan
Abstract
In order to reveal control properties
induced by antagonistic
pairs of bi-articular
muscles, we performed
theoretical simulation analyses as well as actual arm robotic analyses
utilizing two-joint link models equipped
with pneumatic
artificial rubber actuators.
The
previously reported EMG patterns of human movements were well explained from activating level patterns
of the actuators
in terms of mechanical
control engineering.
Results
obtained in the present studies strongly suggest that the existence of the antagonistic pair of
bi-articular
muscles could positively contribute
to the compliant properties
of the multiarticular extremity, and to independently
control position and force at the endpoint of the
extremity, leading to smooth, fine and precise movements.
1. Introduction
We have been studying functional significances of bi-articular muscles mainly from the viewpoint of electromyographic kinesiology - for many
years. Electromyograms CEMGs) recorded from antagonistic pairs of biarticular muscles have shown idiosyncratic patterns. Previously reported
were:
* Corresponding
author.
0167-9457/94/$07.00
0 1994 Elsevier
SSDI 0167.9457(94)00034-4
Science
B.V. All rights reserved
612
M. Kumamoto et al. /Human
Movement Science 13 (1994) 611-634
(1) Reversal of the discharge patterns of the rectus femoris and the
hamstrings prior to heel contact when the upper body was flexed forward
at the hip joint during gait cycles (Kumamoto et al., 1981; Oka, 1984).
(2) A reversal of the discharge patterns of antagonistic pairs of bi-articular muscles of both the upper arm and thigh during extension movements
when a functional force direction was changed under isometric conditions
(Yamashita et al., 1983; Kumamoto et al., 1985).
(3) A reversal of the discharge pattern of the biceps brachii and the
triceps brachii long head recorded during the hand contact period in the
performance of forward handsprings between skilled and unskilled subjects
(Oka et al., 1992).
Such patterns seemed to be controlling output force rather than to be
developing or transmitting propulsive force as had been demonstrated in
the gastrocnemius (Van Ingen Schenau et al., 1987; Van Soest et al., 1993).
As the gastrocnemius has the antagonistic tibialis anterior muscle, but not
acting on the knee joint, our interest was particularly focused on the
existence of the antagonistic pair of bi-articular muscles.
From a mechanical engineering viewpoint, Hogan (1984, 1985a,b) had
earlier suggested that co-contraction of antagonistic bi-articular muscles
could modulate impedance of the multi-joint link system. However, we
need a more detailed analysis in order to understand what happens to the
idiosyncratic EMG patterns of the antagonistic pair of bi-articular muscles
in terms of mechanical control.
In the present paper, we attempt to take a different mechanical engineering approach in order to elucidate mechanical control properties of an
antagonistic pair of bi-articular muscles. In this way it was hoped to be able
to directly explain the presence of the idiosyncratic activity pattern of the
bi-articular muscles observed in most common human movements.
In everyday movements of walking, running, standing up and sitting
down using the legs, opening and closing doors, or pushing and pulling
using the arms, the hip and knee joints simultaneously extend or flex and
the shoulder flexion or extension and elbow extension or flexion simultaneously occur in the arm.
For such common movements, it should be noted that when the proximal
end of a bi-articular muscle synergistically acts on the attached joint, the
distal end of the muscle opposingly acts on the other joint. For example, in
leg extension, the hamstrings (Hm) synergistically act on the hip joint but
opposingly on the extending knee joint, while the rectus femoris (Rf), an
M. Kumamoto et al. /Human
Movement Science 13 (I 994) 61 l-634
613
antagonist of the Hm, synergistically acts on the knee joint but opposingly
on the extending hip joint. Vice versa for leg flexion.
Such an apparently redundant and also contradictive existence of antagonistic pairs of bi-articular muscles is commonly seen in animals as well as
in human beings, but has seldom been encountered in mechanical engineering or robotics.
In order to elucidate mechanical control properties of antagonistic pairs
of bi-articular muscles, we have carried out mechanical engineering model
analyses, i.e., actual arm robotic experiments as well as theoretical simulation analyses.
2. Analyzing protocol
Mechanical link models utilized in the present paper consist of three
segments and two joints as shown in Fig. 1. In Fig. lA, two couples of the
antagonistic mono-articular muscles of fl and el, and of f2 and e2 were
attached to the joints of Jl and 52, respectively. Contrastingly, in Fig. lB, a
couple of the antagonistic bi-articular muscles f3 and e3 were attached to
both joints of Jl and 52 in addition to the mono-articular muscles.
To compare with the EMG patterns recorded during extension of upper
or lower extremity, model postures and analyzing conditions were set the
same as in the EMG experiments. The maximal output force was developed at the endpoint E under isometric conditions, and the force direction
was limited between line Jl-E (Y-axis> and line J2-E, where bi-articular
muscles were functioning contradictively at both their ends and a clear
reversal in EMG patterns of bi-articular muscles was observed.
In order to eliminate gravitational effects, we set up the link models
utilized in both simulation and actual arm robotic studies on a horizontal
desk, because the weight of upper or lower extremity extending in a sagittal
plane could be compensated for by a counterbalance to eliminate gravitational effects during EMG recording.
The visco-elastic muscle model used in the present experiments was that
demonstrated
by Ito and Tsuji (1985), where they also discussed the
length-tension
and force-velocity relation of the contracting skeletal rr uscle shown by Dowben (1980).
According to Ito and Tsuji (1985), if muscular contraction force F is a
function of activation level (Y(0 I (YI l),
+a@,
v),
(1)
614
Fig. I. Two-joint link models empfoyed in the present experiments. fA1 Only mono-articular muscles
are incorporated. (ES)~~-arr~cul~r muscles are inc~~r~ted
in addition to Mona-art~cu~~r muscles.
where g(L, V) is a no~~~~ear function e~r~ssjng the relation between
tension and length 15, and between force and shortening velocity V
~~owbe~, 1980).
Applying Tay~or’s expansion in the ~e~g~~or~ood of L = resting length
I, and shortening velocity Y = 0, and negIecting more than the second-order
terms for sufficiently small f L - I,) and V, we have
==f* -
px - yi,
where:
fO: maximum tension of isometric contraction
(21
CV = 01 at resting length f,,
M. Kumamoto et al. /Human
Movement Science 13 (1994) 61 l-634
615
(A) F = II - kux - buji
(C) T = (Fr - Fe)r
(D) TI = (Ffl -Fel)r + (Fn - Fe3)r
Fig. 2. (A) Visco-elastic
muscle model used in the present experiments.
F: output force, u: contractile
force, k: elastic coefficient,
b: coefficient
of viscosity, X: contracting
length, contracting
direction is
positive. ,k shortening
velocity. (B) Under isometric
condition,
a coefficient
of viscosity could be
discarded,
1= 0. (C) Joint moment T developed by output forces Fr and I$ of a couple of antagonistic
mono-articular
muscles. 0: joint angle, r: radius of joint pulley. (D) Joint moment T, developed by Frs
and F,, of antagonistic
pair of bi-articular
muscles in addition to F, and F, of antagonistic
pair of
mono-articular
muscles.
X:
*.
X.
contracting
contracting
shortening
length of muscle,
direction is positive,
velocity.
a&?
z
y,=P,
v-o
a&?
ag
a z ;Ib”=
,,<O
From Eqs. (1) and (2),
F=u-kux-bui,
where:
k+
b+
0
0
-y.
L - 1, = -x,
616
M. Kumamoto et al./Human
Movement Science 13 (1994) 611-634
Eq. (3) is a visco-elastic muscle model as shown in Fig. 2A. This has the
distinctive feature that visco-elastic coefficients are not constant as is
usually found, but are in proportion to contraction force U.
As human EMG experiments were performed under isometric conditions, all simulation and actual model analyses were attempted under
isometric conditions. For this reason, the coefficient of viscosity of the
muscle model could be discarded - as shown in Fig. 2B.
F=u-kux.
(4)
In Fig. 2C, from Eq. (4) force outputs Ff and Fe would be:
Ff = uf - ku,r8,
Fe = u, + ku,r8,
(5)
where:
8: joint angle,
r: radius of joint pulley.
Joint moment T as shown in Fig. 2C would be:
T= (Ff-F,)Y=
( uf - u,)r - (uf + u,)kr2&
(6)
Joint compliance C caused by Ff and Fe would be expressed as follows:
-1
c=
(7)
(ur + u,)kr2 ’
The relation between a very small change in joint moment of AT and a
very small change in joint angle of A0 would be:
AT = fas.
(8)
If a couple of bi-articular muscles are attached to both joints as shown in
Fig. 2D, joint moment T, would be:
T, = (Fo - F,lP + (F,, - &)y
= (Ufl - u& - (% + %P2&
+ (% - u,&-
- (uf3 + U&r2(~,
+ ‘32).
(9)
Generally, in a two-joint link system such as in Fig. 1, the relationship
between coordinates of endpoint E of the link, (x, y>, and joint angles of
13, and 0, would be expressed as follows:
cos 8, cos(8, + 0,)
(1l
X
Y
=
)
1,
sin f3,, sin(B, + 0,) I( 1,I ’
(10)
M. Kumamoto et al. /Human
Movement Science 13 (1994) 61 l-634
617
where 1, and 1, were lengths of the segments between joints Jl and 52, and
between joint 52 and endpoint E in Fig. 1, respectively.
The relation between very small changes in coordinates of the endpoint
E (Ax, Ay) and very small changes in joint angles (be,, AO,) would be:
-I,
sin 8, - 1, sin(8, + O,), -1, sin(B, + e,)
1, cos 8, + I, c0s(e, + e,), 1, c0s(e, + e,)
The relation between joint moments CT,, T,), and x - y components
the force exerted at endpoint E (F’, F,) would be:
-I,
sin 8, - 1, sin(8, + e,),
1, cos e1 + 1, c0s(e, + e,)
-1,
sin(8, + e,),
1, c0s(e, + e,)
(11)
of
(12)
Consequently, the relation between very small changes in joint moments
(AT,, AT*) and very small changes in the x-y components of the force
exerted at endpoint E (A F,, A F,) would be expressed as follows:
-1, sin 8, - 1, sin(B, + e,),
I, cos 8, + I, c0s(e, + e,)
-1,
1, c0s(e, + e,)
sin(0, + e,),
(13)
3. Output force and muscular activation
patterns
in two joint link models
3.1. Theoretical simulation analyses
3.1.1. Link model equipped only with mono-articular muscles
A link model equipped only with mono-articular muscles is shown in Fig.
1A. In this link model, joint moments of T, around the joint Jl and of T2
618
M. Kumanoto
et al. /Human
Movement Science 13 (1994) 611-634
around the joint 52 would be specified from Eq. (6) as follows:
Tr = (&, - &)r
= (+, - u,&
- (url + u,,)kr24,
(14)
$2)Y
(+2
+
q&r24,
T2 = (4, - &2b=
(42
where:
Fn7 F,1: force outputs developed by the muscles fl and el attached to the
joint Jl, respectively,
force outputs developed by the muscles f2 and e2 attached to the
42,
Fe6
joint 52, respectively,
contractile forces of the muscles fl and el attached to the joint Jl,
Ufl, ue1:
respectively,
__~J_p_J_cT
‘d QL..~.L&__I~._.J.L__l_.._
20
IO
(B)4O’.c-~-.~._r--~d-
A -A
I .
F(N)
A/’
/
AI
/
.A
40
30
-A-
-A-
A-
8fCdeg)
4-A
l()Cj
\
UC%)
‘1.
‘\
.\’
‘\
-
50
Fig. 3. Output
force and muscular
activation
patterns
in the two-joint
link model where only
mono-articular
muscles were incorporated
as shown in Fig. 1A. Abscissa: output force direction
Of
(deg). Ordinate (left): output force F (Newton). Ordinate (right): activation level (contractile
force) U
(%). Triangle marks correspond
to the marks of muscles shown in Fig. lA, A : fl, A : el, v : f2, v : e2.
Closed circle 0: output force. The symbol marks are the results obtained from the robot experiments,
and the broken and solid lines represent the simulation analyses. Panel A: Both pairs of muscles were
activated. Panel B: Only two agonist muscles of fl and e2 were activated.
M. Kumamoto et al. /Human
Mooement Science 13 (1994) 61 l-634
619
contractile forces of the muscles f2 and e2 attached to the joint 52,
respectively.
Since F, and F, could be obtained from Eqs. (12) and (14), Fmax (of)
could be calculated.
In the case of Fig. 3A, the two antagonistic pairs of mono-articular
muscles fl and el, and f2 and e2 were activated, but in the case of Fig. 3B,
only the agonistic mono-articular muscles fl and e2 were activated.
As the contractile force of each muscle is determined by its activation
level, we chose a level for each muscle which would result in the maximal
output force at endpoint E (in both cases A and B) being exerted. The
activation levels chosen are demonstrated with broken lines in panels A
and B.
In the link model with 8, = 45” and 13,= 90”, Fmax values calculated
with change in 0f from 0” to 45” are shown in solid lines in both panels.
Force lines showed the same value and the same changing pattern with the
summit around 18” in both panels.
In Fig. 3A, the agonistic mono-articular muscles fl and e2 were fully
activated throughout changes in of. Changing patterns of the activated
levels of the antagonistic muscles el and f2 in Fig. 3A, and that of the
agonistic muscles fl and e2 in Fig. 3B were a mirror image. This, results in
the same changing pattern for the output force.
UfZ?
ue.2.*
3.1.2. Link model equipped with antagonistic pair of bi-articular muscles
As was shown in Fig. lB, when the antagonistic pair of bi-articular
muscles was incorporated into the two-joint link model in addition to the
mono-articular muscles, joint moments of TI and T2 derived from Eq. (9)
would be as follows:
T, = (Ff, - Felb + (Ff3 - Fe&
=
T2 =
=
(ufl- u& - (Ufl + Ue,)krZq
+ (Uf3- Ud)Y
- (Uf3 + ueJkr*(%+ 4),
(Ff2
-
Fe2b
+
(Ff3
-Fe&
(Uf2
-
42)~
-
bf2
+
-
(Uf3
+
ue3)kr2(8,
ue2P2~2
+ 19,).
+
hf3
-
%)Y
(15)
As was mentioned previously, F max(0f) could be calculated by F, and
from Eqs. (12) and (15) in this model.
F, obtained
620
M. Kumamoto et al. /Human
Movement Science 13 (1994) 61 I-634
Activation levels of all the muscles employed were selected by reference
to the EMG patterns recorded during the leg (Kumamoto et al., 1985) and
arm extensions (Yamashita et al., 1983; Kumamoto, 1992) under isometric
conditions with maximal effort exerted. The agonistic mono-articular muscles were seen to maintain almost fully activated levels during changes in
output force direction, and the antagonistic pair of bi-articular muscles
showed a criss-cross pattern (Fig. 6).
From the results mentioned above, activation levels (contractile forces)
postulated in order to exert the maximal output force at endpoint E were
as follows:
Ufl + U,l = lOO%,
(Ufl = lOO%,
uel = O),
Uf2 + u,z = lOO%,
UF3+ u,3 = 100%.
(z+* = 0,
u,* = lOO%),
(16)
Activated levels of the bi-articular muscles are shown in thin solid lines
in Fig. 4. For Fig. 4A, 8, = 30” and 8, = 120”; for Fig. 4B, 8, = 45” and
8, = 90”; and for Fig. 4C, 8, = 60” and 0, = 60”.
Fmax values calculated with changes in 0f are shown in bold solid lines
in Fig. 4.
3.2. Robot arm experiments
Pneumatic artificial rubber actuators (PRA, Bridgestone Co. RUB-515S),
were installed on the two joint link model via sprockets and chains as
shown in Fig. 5. Rotary encoders were fitted at the joints Jl and 52, and
joint angles o1 and 8, were measured. An L-shaped force detector,
attached with strain gauges, was set at endpoint E of the robot arm, so that
x - y components of the force exerted at endpoint E could be measured,
allowing F max and 0f to be calculated.
F max and Of values were obtained under the same experimental
conditions postulated for the simulation analyses, and are plotted in Figs. 3
and 4 as shown by symbol marks.
As is obvious from Figs. 3 and 4, there was full coincidence between the
actually measured values and the simulated results.
3.3. Discussion
As demonstrated
pair of bi-articular
in this investigation, the existence of the antagonistic
muscles could produce a smoothly changing output
M. Kumamoto et al. /Human
1
0
IO
Movement Science 13 (1994) 61 I-634
I
I
I
20
30
40
__-__-__*__--__t_____-__
(B) 40
621
I
Bftdeg)
IO0
F(N)
U(7r
20
50
0
IO
20
30
40
t3f (dep)
0
IO
20
30
40
8f (deg)
Fig. 4. Output force and muscular
activation
patterns
in the two-joint link model where a pair of
bi-articular
muscles in addition
to mono-articular
muscles were incorporated
as shown in Fig. 1B.
Abscissa: output force direction
.9f (deg). Ordinate
(left): output force F (Newton). Ordinate
(right):
activation level (contractile
force) U (o/o). Triangles
A : muscle fl and A : el, and squares
0 : muscle f3
and n : e3. Closed circle 0: output force. The symbol marks are the results obtained from the robot
experiments,
and the broken and solid lines are from the simulation
analyses. Panel A: 8, = 30” and
0, = 120”; panel B: 0, = 45” and e2 = 90”; and panel C: f?r = 60” and 0, = 60”.
622
M. Kumamoto
et al. /Human
Movement
Science 13 (1994) 611-634
Rubber Actuator
Fig. 5. An arrangement
of pneumatic
artificial rubber actuators
on the robot arm. A pair of the
actuators of fl and el, and a pair of f2 and e2 were attached to the joints Jl and 52 with a chain and a
sprocket, respectively, so that they could act as mono-articular
muscles, whereas a pair of f3 and e3 was
attached to both joints Jl and 52 as bi-articular
muscles.
force curve with change in the force direction (Fig. 41, while the monoarticular muscles alone, i.e. without the bi-articular muscles, could not
produce such a smooth curve (Fig. 3). The mathematical model, previously
proposed for the two-joint link system (Kumamoto, 1984, 19921, could not
develop such a smooth force curve, as the model did not essentially involve
the bi-articular function.
However, the integrated EMG patterns including the bi-articular muscles which were recorded during arm extension (Yamashita et al., 1983;
Kumamoto, 1992) and leg extension (Kumamoto et al., 1985) were quite
similar to the results obtained in this investigation. The EMGs of the leg
extension are presented in Fig. 6, where the posture was equivalent to the
case of Fig. 4B. As shown in Fig. 6, the Vm of the mono-articular knee
extensor kept almost full activity level throughout changes in the force
direction from K, to H,,, and the Rf and the MH of the antagonistic pair
of the bi-articular muscles showed a criss-cross pattern. K, and H,
correspond to of = 0” (Y axis) and of = 45”, and Rf and MH correspond to
e3 and f3 in Figs. 1B and 4B, respectively. Their EMG patterns are
essentially the same as observed in Fig. 4B.
The more the posture extended, the sharper the criss-cross pattern of
the bi-articular muscles, and the larger the output force as shown in Fig. 4.
Such a phenomenon will be discussed in the following sections in terms of
control engineering.
M. Kumamoto et al. /Human
0
0
0
A
A
**---______
--__ *_
-_,z
I
‘Ko
-3
I
623
Movement Science 13 (1994) 611-634
_-
Vm
VI
Rf
Tfl
Gm
___a
I
‘K2
‘C
‘H2
‘HI
‘Ho
100%
50
0
Fig. 6. Changes
in integrated
EMGs with change in functional
force direction
under isometric
conditions
with maximal efforts. The experimental
posture
was equivalent
to Fig. 3B. Abscissa:
proportional
positions where the functional
force lines crossed the thigh between the knee and hip
joints. K, and HO correspond
to Sf = 0” (Y-axis) and 0f = 45” in Figs. 1B and 4B, respectively.
Ordinate:
normalized
integrated
EMG (o/o). Vm: Vastus medialis, Vl: Vastus lateralis,
Rf: Rectus
femoris, Tfl: Tensor fascia lata, Gm: Gluteus maximus, MH: Medial hamstrings.
Rf and MH correspond to e3 and f3 in Figs. 1B and 4B, respectively. (Fig. 2, of Kumamoto
et al., 1985, reproduced
with
permission of the authors and the publisher.)
4. Compliant
properties
of two-joint models
In the two-joint model equipped only with mono-articular muscles,
compliance C, of the joint Jl which was caused by the mono-articular
muscles fl and el, and compliance C, of the joint 52, caused by the
mono-articular muscles f2 and e2, could be derived from Eq. (7) as follows:
-1
c, =
GGl + ue,)kr2 ’
-1
c, =
(Uf2
+ Ue2)k?
*
(17)
Relations between very small changes in the joint moments of AT, and
AT2, and very small changes in the joint angles of AOr and A8, were
624
M. Kumamoto
et al. /Human
Mooement Science 13 (1994) 611-634
derived from Eq. (8) as follows:
AT, = $A,,,
1
AT, = fas,.
2
Relations between very small changes in the coordinates of endpoint E,
(Ax, Ay), and very small changes in the x -y components of the force
exerted at endpoint E, (AF’, AF,), could be calculated from Eqs. (111, (13)
and (18) as follows:
(z)
= [::::
(19)
:::)
where:
a,, = {II sin 8, + 1, sin(8, + 82)}2C, + {I, sin(8, + 82)}2C2,
a12= a21
= [-I, sin 8,(1, cos 8, + 1, cos(0, + 0,))
-1, sin(0, + e,){z, cos 81 + I, c0s(el + e,)}]c,
- 1; sin(8, + e,) c0s(e, + e,)c,,
a22 = (11 cos
8, + 1, c0s(el + e2)}2c, + {I, c0s(e, + e2)j2c2.
(20)
Now, in the two-joint link model equipped with bi-articular muscles, in
addition to the mono-articular muscles, existence of the bi-articular muscles f3 and e3 might provide additional influences on both joint compliantes of C, of Jl and of C, of 52. C, and C, were shown in Eq. (17).
Compliance C, caused by the bi-articular muscles f3 and e3 can be derived
from Eq. (7) as follows:
(21)
Relations between very small changes in joint moments of AT, and AT,
and very small changes in the joint angles of AOr and A8, can be derived
from Eq. (8) as follows:
AT, = ;A&
1
+ ;(A8,
3
+ A&),
AT2 = ;dB,
2
+ ;(A8,
+ be,).
3
(22)
M. Kumamoto
et al. /Human
Movement
Science 13 (1994) 61 l-634
625
Relations between very small changes in the coordinates of endpoint E
(Ax, Ay) and very small changes in the x - y components of the force
exerted at endpoint E (AF,, AF,,) can be derived from Eqs. (ll), (13) and
(22) as follows:
[z]
= [z:;:
I:]
(23)
[z$
where:
aI, = {II sin 8, + 1, sin(B, + 82)}2C,
+ 21, sin(B, + O,){Z, sin O1+ I, sin(8, + B,)}C,
+ 1; sin(8, + 82)2C,,
a12= a21
= -{Z1sin O1+l, sin(0, + e2)}{zl
cos 8,+I, c0s(e,+ 6,)}C,
- [4 cos(% + e2)vlsin 8,+ 1, sin(8, + e,))
case,+I, c0s(e,+e,>)]c,
+I, sin(8, + e2){1,
-1; sin(B, + e,)c0s(e,+ e2)cC,
a22= {11cos 8,+I, c0s(e,+ e2)}2c,
+ 21, c0s(e,f e,){z,
cos 8,+I, c0s(e,+ e,)}c,
+l; c0s(e,+e212cC,
(24)
in which:
c, =
Cl(C2+ CJ
c,=
c,+c,+c,'
-c,c2
c,+c,+c,'
c,=
c2G+cJ
c, +c,+c,
. (25)
Since relations between very small changes in the x - y components of
the force exerted at endpoint E ( AF,, A F,) and very small changes in the
coordinates of endpoint E (Ax, Ay) can be derived from Eq. (19) or (23) as
follows:
/
w\
I
=
alla22
-
2
al2
’
alla22
-a12
4
\ alla22
/
-a12
a22
Ax
-
42
(26)
a11
-
2 ’
a12
alla22
-
AY
42
I \
M. Kumamoto et al. /Human
626
Mocement Science 13 (1994) 611-634
Fig. 7. Effects of the existence of the bi-articular
muscles on stiffness
two-joint link model. Results from three postures, 0, = 30” and 0, =
0, = 60” and 0, = 60”, were superimposed
in each panel. Panel A: The
of the bi-articular
muscles as well as mono-articular
muscles; panel B:
muscles without the bi-articular
muscle, panel C: only with agonistic
joint. For further explanation,
see the text.
The potential
energy
Ep at endpoint
I
a22
4la22 EP = (x, Y)
E would be:
alla22
-a12
, alla,,
\’
-a12
2 ’
a12
-
’
2
a12
x
(27)
a11
-
2 ’
a12
control at the endpoint of the
120°, 0t = 45” and 0a = 90”, and
model was equipped with a pair
with two pairs of mono-articular
mono-articular
muscle on each
alla22
-
2
a12 /
,
y
,
M. Kumamoto et al. /Human
Movement Science 13 (1994) 611-634
621
Eq. (27) can be rewritten as an elliptical equation as follows:
a22
( a11422
-
a11
X2+
af2)Ep
( alla22
-a?,)&
Y2
Thus, equipotential energy lines were elliptical in shape. When al,, al2 and
a22 of Eq. (28) were substituted by those of Eq. (201, equipotential energy
lines of the link model with the mono-articular muscles (Fig. 1A) can be
drawn as shown in Figs. 7B and 7C. In Fig. 7B, two pairs of antagonistic
mono-articular muscles were utilized, and in Fig. 7C, only the agonistic
mono-articular muscles.
In the model equipped only with the mono-articular muscles, the direction 0f of output force Fmax of the model can be changed by changes in
activation levels of ufl, u,i, uf2 and u,*. Therefore, change in 0f might
cause changes in compliance C, and C, as was shown in Eq. (17). That is,
changes in elements of matrix (19) might further result in changes in the
elliptical shapes, as was shown in Figs. 7B and 7C.
In the model equipped with the bi-articular muscles as well as monoarticular muscles, when Eq. (28) was substituted by Eq. (241, equipotential
energy lines of the model could be drawn as shown in Fig. 7A.
In each figure, the lines of different postures with 8, = 30” and 8, = 120”,
8, = 45” and 8, = 90”, and 8, = 60” and 8, = 60” were superimposed.
The experimental conditions postulated were shown in Eq. (161, in Eqs.
(17) and (211, compliances Cl, C, and C, were constant. That is, no change
in elements of matrix (27) resulted in constant elliptical shape independent
of of, as was shown in Fig. 7A. Direction of force output (of> was
dependent on the ratio of u, and u,~.
5. Hybrid position/force
muscles
control
by an antagonistic
pair of bi-articular
In the two-joint link model, where only the mono-articular muscles were
present, stiffness of the link model and output force direction were not
controlled independently.
628
M. Kumamoto et al. /Human
Mooement Science 13 (1994) 611-634
However, when the link model had an antagonistic pair of bi-articular
muscles in addition to the mono-articular muscles, stiffness of the link
model and output force direction were controlled independently.
Now, the effects of the existence of an antagonistic pair of bi-articular
muscles on independence of position (displacement) and force exerted at
the endpoint of the two joint link model are examined.
5. I. Simulation study
The length of the segments of the link model used (Fig. 1) is:
(29)
1, = I,.
From the experimental conditions allowed in the link model, the relationship between joint angles 0r and e2 is:
8, =
T
-
(30)
28,.
When the two-joint link model has only mono-articular
of Eq. (20) would be as follows:
a12 =
muscles, uI2 = u2r
(31)
a,, = 1; sin 0, cos O,C,.
Since C, f 0, except in an extreme case such as 8, = 0 (0, = r> or
8r = n-/2(e2 = O), matrix (19) was not a diagonal matrix. Therefore, relations between very small changes in force and position exerted at the
endpoint of the two-joint link model could be derived from Eq. (26) as
follows:
a22
AF, =
a1la22
AF,=
-
2
-
a12
a12
a12
Axalla22
2
Ax+
-
2
a12
AY,
a11
AY.
(32)
alla22
- 42
a1 la22 - a12
Thus, in the two-joint link model with only mono-articular muscles,
under general experimental joint angle conditions, position (displacement)
and force exerted at the endpoint of the two-joint link model could not be
controlled independently.
On the other hand, when the two-joint link model has an antagonistic
pair of bi-articular muscles, in addition to the mono-articular muscles, from
postulated conditions of the muscular forces (161, relations among compliantes C,, C, and C, would be:
Cl =
c, = c,.
(33)
M. Kumamoto et al. /Human
Therefore,
Movement Science 13 (1994) 61 I-634
629
from Eq. (25):
c, = -+c,.
c,=c,,
From the relations of (34) and the experimental
(301, uI2 = u2, of Eq. (24) would be as follows:
aI2 = C.221
= 0.
(34)
conditions of (161, (29) and
(35)
This would lead matrix (231, under the joint angular condition of (301, to be
a diagonal matrix.
Therefore, relations between very small changes in force and position
exerted at the endpoint of the two-joint link model would be predicted
from Eq. (26) as follows:
AF, = LAX,
a11
AFY = IAy.
a22
(36)
From the results mentioned above, it could be concluded that, in the
two-joint link model, the existence of the antagonistic pair of bi-articular
muscles contributes to control position and force exerted at the endpoint of
the link model independently.
5.2. Actual link model experiments
The link model used in this investigation was the same as that used in
Section 3.2 (Fig. 5). In order to make very small changes in displacement
along the Y-axis, Ay, we mounted the L shape force detector on a fine
manipulator, so that any change in force exerted on the X-axis, AF,
induced with Ay could be detected.
Now, measured values of AF, induced by Ay in the two-joint link
models with or without the bi-articular muscles were plotted in open and
closed circles as shown in Fig. 8, respectively.
As shown here, there was no change in AF, in the link model with
bi-articular muscles, whereas, in the link model with only the mono-articular muscles, AF, increased with increase in Ay.
From the simulation studies mentioned previously (see Section 5.1), the
expected AF, values in the two-joint link models with or without the
bi-articular muscles were derived from Eqs. (36) and (32), and are demonstrated in horizontal and diagonal bold solid lines as shown in Fig. 8.
630
M. Kumamoto et al. /Human
A
-10
n
_
o
Mooement Science 13 (1994) 611-634
Y (mm)
-5
0
0
Fig. 8. Effects of the existence of the antagonistic
pair of bi-articular
muscles on hybrid position/force
control at the endpoint of the two-joint link model. Changes in output forces exerted at endpoint E of
the mode1 along the X-axis (AF, (Newton); ordinate) with very small changes in displacement
of the
endpoint
along the Y-axis (Ay (mm); abscissa) were plotted. Results obtained
from the simulation
study are demonstrated
in bold solid lines, and results obtained from the actual robot arm experiment,
by the following symbols: open circle: the model with the antagonistic
pair of bi-articular
muscles as
well as the mono-articular
muscles; closed circle: only the mono-articular
muscles. It was obvious that,
when the model was equipped with the antagonistic
pair of bi-articular
muscles, there was no change in
AF, with changes in Ay.
The results obtained from the robot arm experiments perfectly coincided
with the results obtained from the simulation studies.
It can be concluded
that the existence
of the antagonistic
pair of
bi-articular
muscles gives rise to fine, smooth and precise movement
patterns characteristic
if both human and animal motion - characteristics
that are not observed in even sophisticated
modern robots.
6. General discussion
Effects of the existence of an antagonistic pair of bi-articular muscles on
the equipotential
energy lines have been clearly demonstrated
- Fig.
7A-C. The existence of an antagonistic pair of bi-articular muscles resulted
in constant stiffness, which is a reversal of equipotential
energy, curves
independent
of c9f as shown in panel A. On the other hand, when the
model incorporated
only mono-articular
agonist muscles, the stiffness curves
changed their shapes and directions with change in 19f as shown in panel C.
Existence of the antagonistic pair of mono-articular
muscles made fluctua-
M. Kumamoto et al. /Human
Movement Science 13 (1994) 61 l-634
631
tions of the stiffness curves with changes in of smaller than those of panel
C, as shown in panel B.
Thus, the existence of an antagonistic pair of bi-articular muscles contributes to stable stiffness response against disturbances from any direction,
whereas the existence of antagonistic mono-articular
muscles, without
bi-articular muscles, does not show such a stable response.
Since stiffness is a reversal of equipotential energy, the maximal stiffness
exerted at the endpoint of the link appeared at of = 0, where the direction
was passing through joint Jl, as shown in Fig. 7A. The more extending the
posture, the larger the stiffness exerted at the endpoint of the link, also
shown in Fig. 7A. This tendency was accompanied by a sharper change in
the crisscross pattern of the antagonistic pair of bi-articular muscles as
shown in Fig. 4.
From the results mentioned above, it can be concluded that the existence of an antagonistic pair of bi-articular muscles contributes to stiffness
control of the endpoint of the limb(s) at foot contact in adult gait
(Kumamoto et al., 1981; Oka, 19841, in infant gait (Okamoto et al., 19831,
and during hand contact in the handspring (Oka et al., 1992).
Further, the discharge pattern of the Rf and the HM reverses prior to
the first step after the upper body is suddenly flexed at the hip joint during
the swing phase of a gait cycle, probably within 30 ms (Oka, 1984). Postural
change in the hind limb induces reversed electrical activity of the semitendinosus in the decorticated rabbit (Vidal et al., 1979). Therefore,
peripheral innervation will be necessary on the bi-articular muscles contributing to stiffness control, but a central nervous command will not be
necessary. It might allow a quick response against perturbation during gait
cycles or sport activities.
The fact that the discharge pattern of the bi-articular muscles has
reversed prior to the first step after hip joint flexion during the gait swing
phase (Oka, 1984), suggests that joint angular information of the hip,
without muscular tension information, could give rise to such a reversed
discharge pattern. On the other hand, in leg extension (Kumamoto et al.,
1985) or in arm extension (Yamashita et al., 1983), as the experiments were
performed under isometric conditions, muscular tension and joint pressure
information without joint angular information could result in the reversed
discharge patterns in the bi-articular muscles.
It is quite difficult to elucidate a substantial difference between control
properties of limbs with or without bi-articular muscles in human beings or
even in animals. However, as regards the stiffness control of the limb, the
632
M. Kumamoto et al. /Human
Mouement Science 13 (1994) 611-634
results obtained from the link model analyses demonstrated perfect coincidence with the results obtained from human subjects. Indeed, relations
between the discharge patterns of antagonistic pairs of bi-articular muscles
and the reaction forces or postural changes could be explained from the
results obtained by the simulation analyses and the actual robot arm
experiments. Therefore, although direct evidence could not be obtained, it
seems natural to infer that the existence of antagonistic pairs of bi-articular
muscles contributes to the independent
control of position and force
exerted at the endpoint of limbs in human beings and animals resulting in
smooth, fine and precise movements.
The results obtained from the present analyses indicate that the stiffness
control or independent position/force control can proceed in the presence
of an antagonistic pair of bi-articular muscles, without feedback by environmental constraints, as was suggested by Hogan (1984, 1985a). Existence of
bi-articular muscles might allow controlled movements even in deafferented monkeys (Taub et al., 1975).
The unique functions of the bi-articular muscles can be summarized into
two categories, i.e., propulsive force transmission/production
and control
properties. The function of propulsive force transmission was discussed by
Van Ingen Schenau and colleagues (Van Ingen Schenau et al., 1987; Van
Soest et al. 1993). The hamstring muscles and the gastrocnemius are
functioning effectively to transmit propulsive forces produced by the bulk
trunk muscles to the feet. The hamstrings have an antagonist on the
opposite side, but the gastrocnemius has no antagonist to act at the knee
on the front of the lower leg. Even in the lower leg of lesser apes, which
use their hind limbs to grasp a branch of a tree, the gastrocnemius exists by
itself without an antagonist on the opposite side. Probably, the lower leg of
the hind limb of mammals might well develop to transmit propulsive force
as in hoofed animals, losing an antagonist to the gastrocnemius. Existence
of an antagonistic pair of bi-articular muscles on the upper leg will be
necessary for precise control of position and force exerted at the foot. The
rectus femoris, an antagonist of the hamstrings, has a very small insertion
area, thus, its function might be control rather than force transmission.
Acknowledgements
The authors are gratefully indebted to Shuichi Koyama, Associate Professor, Kansai Medical University, for his helpful discussions during prepa-
M. Kumamoto et al. /Human
Movement Science 13 (1994) 611-634
633
ration of this paper. This work was supported by the Grant-in-Aid for
Scientific Research on Priority Area: “Biomechanics”, No. 04237222, from
The Ministry of Education, Science and Culture, Japan.
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