21
COMMUNICATIONS
A Physiologically Based Model of Neuromuscular
System Dynamics
Abstract-A model of the neuromuscular system based on the physiology of muscle is described. All elements of the model have been discussed in earlier literature but certain new implications are considered here. THE MODEL The steady-state force-velocity curve for muscle (both human and isolated) is well established[2]-[6] to be of the form shown in Fig. 1. The data are well fitted by a family of hyperbolas:
F = zFo
(1)
I
where z is the percent stimulation (ratio of actual stimulation to maximum; 0 < z < 1), VO is the maximum velocity, Fo is the maximum isometric force, and n is a constant. Bigland and Lippold[51 showed that the stimulation z is proportional to the electromyograph (EMG), at least for steady motion. The force-velocity curve is not as well verified for lengthening muscle, but this relation appears to hold satisfactorily up to the maximum allowable force, which may be twice Fo. The interesting point about this relation is that the stimulation z is multiplicative rather than additive, so that the stimulation in effect controls the damping as well as the isometric force. This is a more important nonlinearity than the fact that the lines of constant stimulation are curved in Fig. 1. In fact, we can linearize this property of (1) to yield
F
zF[l - (1 + 1/n)V/V0].
=
If we assume that the effect of increasing the stimulation is to increase the active cross section of the muscle,['1 we can relate the spring constant to the average stimulation 2. Consider an elastic element with modulus E:
FIAL
=
K
K
=
Koz.
EA = EAo2 /o Lo (3)
This result agrees with measurements made by Wilkie E61 who found the relation between K and the tension level in the muscle shown in Fig. 3. We can neglect, for purposes of this model at least, the variation of isometric torque with angle 0. This variation is not large for the forearm because the effects of variation with muscle length tend to be canceled by the changing lever arm. If we now put two such identical muscles together with a limb (Fig. 4), the equations of motion become
V1 =
d6+Kd2
V2
-do + Kdz2
=
T,
T, = Toz(l - f3VI/Vo) T2 Toz2(1 - 13V2/1VO) Md + T - T2= 1 =
(2)
This F-V relation is valid only for constant velocity; thus to extend the model to cover general motions, a spring must be introduced in series with this steady-state model (see Fig. 2).
ELLo
F/A
(4)
3 a£ I'll
_E
KFo
2,
c 10
NZ= 1.0
Fig. 3. Effective spring constant of biceps versus muscle tension, both measured at the hand. (From Wilkie.l61)
P' ,
h0 0
0
2
4
6
Velocity (m/s)
I
Fig. 1. Biceps force versus steady-state velocity of shortening, both measured at the hand. (From Hill,131. 4] Wilkie,[61 and Bigland and Lippold. [6) I
F0 ] /l
z
Muscle 2
K0
Muscle 1
szB(v)
Fig. 2. Schematic model of a single muscle. Manuscript received August 1, 1967; revised December 7, 1967. This work supported by NSF and NASA traineeships and Grant NsG 107-61.
was
Fig. 4. Schematic of the limb and muscle system.
22
IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, MARCH
where d is the lever arm, T, is the torque exerted by the ith muscle (Ti = Fid), I is the moment of inertia of the limb, the constant p = 1 + 1/n, and Md is the disturbance moment. We can nondimensionalize these equations by defining Ali = v; =
T/To
3VJ/Vo ,Bd dO VO dt
Vo dKo ToT
Note that the difference between stimulations (zl - Z2) appears as a normal linear input, while the sum (zl + Z2) acts as a parameter. Thus the dynamic properties of the neuromuscular system can be changed simply by changing the level of stimulation. The effect of the (2122) term in (7) is to counteract any steady disturbance moment, and if there is none, then this term must be zero. Since we regard (z1 + Z2) as being constant, we can Laplace transform (7). It is convenient to introduce two new variables: a
= ~z1 -
2
cr
21 +
12.
Note that a and a- are restricted to the range shown in Fig. 5. Transforming (7) with Q, D, A, and Thd representing the Laplace transform of w(r), +(r), 6(r), and md()r, respectively, we obtain
t(@d,=yT° Rp _ IV dK 7=
s1s=(s) + (s+ 1)d( s s (Rs'+ Rs+ o-) md =
Then
1 dp, z1 dr 1 dgi2
= =A) + _ (
p1=
2S
2 dr
Z1(1 -V1
82= z2(1 ->2 Rd2 +Rd +(zl +ZCO + md +MdlT-o.2 = R -
(6)
In order to solve these equations we must further linearize them by defining z;(r) = z; + zj(r), where z; is again the average stimulation and zt iS the time varying portion. If z'/z «
