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”=

,,, 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.

References Dowben, R.M., 1980. ‘Contractility’. In: V.B. Mountcastle (Ed.), Medical physiology (14th ed., p. 90). Saint Louis: Mosby. Hogan, N., 1984. Adaptive control of mechanical impedance by coactivation of antagonist muscles. IEEE Transaction on Automatic Control AC-29 (8), 681-690. Hogan, N., 1985a. Impedance control: An approach to manipulation: Part II - Implementation. Journal of Dynamic Systems, Measurement, and Control 107, 8-16. Hogan, N., 1985b. The mechanics of multi-joint posture and movement control. Biological Cybernetics 52, 315-331. Ito, K. and T. Tsuji, 1985. The bilinear characteristics of muscle-skeleto motor system and the application to prosthesis control. The Transactions of the Electrical Engineers of Japan 105-C (lo), 201-208. Kumamoto, M., 1984. ‘Antagonistic inhibition exerted between biarticular leg muscles during simultaneous hip and knee extension movement’. In: M. Kumamoto (Ed.), Neural and mechanical control of movement (pp. 113-122). Kyoto: Yamaguchi Shoten. Kumamoto, M., 1992. Existence of antagonistic bi-articular muscles: Models and EMG studies in man. Abstracts of VIII meeting of The European Society of Biomechanics (pp. 248-249). Kumamoto, M., H. Oka, 0. Kameyama, T. Okamoto, M. Yoshizawa and L. Horn, 1981. ‘Possible existence of antagonistic inhibition in double-joint leg muscles during a normal gait cycle’. In: A. Morecki, K. Fidelus, K. Kedzior and A. Wit (Eds.), Biomechanics (Vol. VII-B, pp. 157-162). Baltimore: University Park Press. Kumamoto, M., N. Yamashita, H. Maruyama, N. Kazai, Y. Tokuhara and F. Hashimoto, 1985. ‘Electrical discharge patterns of leg muscles reflecting dynamic features during simultaneous hip and knee extension movements’. In: D.A. Winter and R.W. Norman (Eds.1, Biomechanics (Vol. IX-A, pp. 324-329). Champaign, IL: Human Kinetics Publishers. Oka, H., 1984. Electromyographic study on lower limb muscle activities during normal gait cycle. Journal of Kansai Medical University 36, 131-152. Oka, H., A. Furuta, M. Yoshizawa and M. Kumamoto, 1992. Antagonistic bi-articular muscles functioning in front handsprings in tumbling and vaulting. Abstracts of VIII Meeting of The European Society of Biomechanics (p. 253). Okamoto, T., Y. Goto, H. Maruyama, N. Kazai, H. Nakagawa, H. Oka and M. Kumamoto, 1983. ‘Electromyographic study of the bifunctional leg muscles during the learning process in infant walking’. In: H. Matsui and K. Kobayashi (Eds.), Biomechanics (Vol. VIII-A, pp. 419-422). Champaign, IL: Human Kinetics Publishers. Taub, E., I.A. Golberg and P. Taub, 1975. Deafferentation in monkeys: Pointing at a target without visual feedback. Experimental Neurology 46, 178-186. Van Ingen Schenau, G.J., M.F. Bobbert and A.H. Rozendal, 1987. The unique action of bi-articular muscles in complex movements. Journal of Anatomy 155, 1-5. Van Soest, A.J., A.L. Schwab, M.F. Bobbert and G.J. van Ingen Schenau, 1993. The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping achievement. Journal of Biomechanits 26, l-8.

634

M. Kumamoto et al. /Human

Motiement Science 13 (1994) 611-634

Vidal, C., D. Viala and P. Buser, 1979. Central locomotor programming in the rabbit. Brain Research 168, 57-73. Yamashita, N., M. Kumamoto, Y. Tokuhara and F. Hashimoto, 1983. Relation between mechanisms of force generation and muscular activity in the movement of upper extremity. Japanese Journal of Sports Sciences 2, 318-324.