Optimal control of antagonistic muscles - Research

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Biological Cybernetics

Biol. Cybern.48, 91-99 (1983)

9 Springer-Verlag 1983

Optimal Control of Antagonistic Muscles* M. N. O~uzt/Sreli and R. B. Stein Departments of Mathematicsand Physiology,Universityof Alberta, Edmonton, Canada

Abstract. Recently, a model for a pair of antogonistic

muscles has been studied (O~uzt/Sreli and Stein, 1982). In the present paper we formulate and investigate the minimization of the costs associated with the time to complete the movement, the oscillation about the endpoint, the energy costs to the muscles to complete the movement, the cost to the nervous system to supply the inputs, and the cost of reliability in the face of perturbing forces. To solve these optimization problems the maximum principle of Pontryagin is employed. In all of these optimization problems, except the energy optimal problem, the optimal controls (active states or nervous inputs) are of the bang-bang type.

I Introduction In a previous paper (O~uzt6reli and Stein, 1982), we

considered a model containing two antagonistic muscles. Muscles generally occur in antagonistic pairs, and one can consider how such pairs of muscles can best be controlled. Control can be exercised through activating each of the muscles separately or through some degree of cocontraction. The problem of optimal control is more difficult for such a natural, biological system than for many man-made systems, since the criteria for optimal control is not precisely known and may change with time in order to adapt the system to changing environmental conditions. Surprisingly little attention has been given to the problem of optimal control in muscular systems (cf. reviews by Margaria, 1976 ; Hatze, 1980). * This work was partly supported by the Natural Sciences and EngineeringResearchCouncilof Canada Grant NSERC-A4345and by the MedicalResearch Councilof Canada Grant MRC-MA-3307 through the Universityof Alberta

In general, one can associate with movements from an initial position to a final position various costs, such as 1) the time t to complete the movement; 2) the oscillation about the end-pont; 3) the energy cost to muscles to complete the movement; 4) the cost to the nervous system to supply the inputs ; 5) the cost of reliability in the face of perturbing forces. In the next section of this paper we will formulate the optimization problem more precisely. Then, we will consider strategies which minimize each of the costs individually, and finally consider the more general situation where a weighted sum of these costs would be nainimized. The weighting will depend on the environmental conditions and the nature of the movement to be completed. To minimize each cost individually, the maximum principle of Pontryagin will be applied. The Hamiltonian and the adjoint system which result in each case is a different type, so that the results have to be derived separately in Sect. IV-VIII. Many details of lengthy, but rather conventional derivations have been omitted to limit the length of the paper. The results, except when energy is being minimized, indicate that the optimal control will be of the bang-bang type, in which one muscle and then the second muscle should be activated maximally. The notation in this paper will be identical to that of the previous one (O~uzt6reli and Stein, 1982), and we will refer to various equations derived in that paper by using a superscript *, rather than repeating the notation and necessary equations here. II Formulation of the Optimization Problem

In making a movement from u = h o to u = h I, one might make the neutral input nl(t ) sufficiently large to

92 = the rate at which energy is being stored in the parallel elastic element,

gJ

E13 = d ( 1 K i 1[Ul(t) -- u(t)] 2) hj

= Ktl [ul(t ) - u(t)] [/il(t )-/i(t)] = rate at which energy is being stored in the internal series elastic element,

S.~'

V" O tt t Fig. 1. Differenttrajectoriesfor movingto a position u = h~ with and without a time constraint

El4 = B j ~ ( t ) = rate at which energy is being dissipated in the viscous element. Hence, gl(t) = E1 o + E11at(t) + Kvlul(t)ul(t)

+ K. [ul(t)- u(t)] [zil(t)- ~i(t)] + B 1/r 5

El.o + E1 ial(t)

--

Ktl [ui(t)

-

u(t)]il(t)

+ [ - gtlu(t ) + (Kil + Kpl)ul(t) + Biiq(t)]iq(t).

E]

/ ' /

Z O O :>O n.,' bJ Z bJ

Since the coefficient of/q(t) is equal to al(t) by virtue of the first equation in (11.3)*, we have

SHORTENING HEAT

~i(t) = E l o + E1 lal(t) M A I N T E N A N C E HEAT

+ al (t)~il(t)

--

Ktl [Ul(/)

--

U(t)]/i(t).

(11.2)

Similarly, for the muscle 2, we have REST HEAT

g2(t) = E20 + E21 a2(t) + Kp2u2(t)i~2(t) VELOCITY Fig. 2. Energy consumption at different velocities of movement can be divided into a number of components. Further details in the text

saturate the active state as(t ) of the first muscle. A minimum time T = t 1 - t o might be achieved via the pathway H 1 in this case (see Fig. 2), where U(to)=h o, u(ts)=hl, and u(t)~:h 1 for to < t < t 1. However, this could only be achieved at a cost in oscillation about the end-point where this is defined by the integral ~1 = ~ {[u(t)- hi] 2 + fi2(t)}dt"

(11.2)

tl

Pathway/-/2 would make this integral zero, but at a cost of infinite time : t s = oo. In going from ut(0)=0 to us(t ) the muscle 1 will consume energy at a rate el(t)=Eio+Elsal(t)+Ela+E13+El,, where Elo = t h e rate of energy expended at rest, E l l = t h e coefficient for the extra rate of energy expenditure required tO maintain the active state a l, d 1

2

El 2 = -~ (sKpiudt))

= Kplul(t)iq(t )

+ Kt2 [u2(t ) - u(t)] [/i2(t) - it(t)] + B2/i22(t) = E20 + E21a2(t) - a2(t)il2(t ) - Kt 2 [ua(t) - u(0]/r

01.3)

Further, energy also will be (i) stored in the external spring at a rate

Ee= Keu(t)u(t), (it) dissipated in the external dashpot at a rate 8 a = D/i2(t), (iii) converted into the kinetic energy of the mass at a rate ~m = M/t(t)a(t) 9 According to the third equation in (II.3)*, the external load will consume energy at a rate

go(t)=ge+~a+g,~ = {f(t) + Kit [u 1( t ) - u(t)] + Ki2 [u2(t)- u(t)] }/fit),

(22.4) and the total rate of energy expenditure is

e(t) = ~o(t) + el(t) + ~2(t) = (El o + E2o) + E11 al (t) + E 21 a2(t) + al(t)iq(t ) -- a2(t)i~2(t ) +f(t)i~(t).

(11.5)

93 Note that the negative sign for a2(t ) arises from the fact that it was defined as a force acting in the negative (i.e. downward) direction in Fig. 1". In his classical experiments Hill (1938) allowed a single muscle to contract against an isotomic lever If(t) was constant] which has negligible mass, viscosity and stiffness. In this case the force will rapidly change the length of the series elastic component (K, ~) and then begin to stretch the viscous element B at a rate which depends on f(t). Under these conditions the terms in the energy equation have a particularly simple interpretation as shown in Fig. 2. There is a resting release of heat Elo which is increased to maintain an active state by the amount El la~(t ). When the muscle shortens, additional heat will be released, al(t)iq(t), and once the series elastic element has been stretched out,/q(t)=/fft). The final term f(t)i~(t) is just the rate at which work is being done, which will be zero at the two extremes when the load is not moving, u(t)= 0, or the muscle shortens at its maximum velocity against no load. For more recent, detailed reviews of muscle energetics, see Mommaerts (1969) or Huxley (1974). Now let, tl

if2 = ~ ~ dt"

(11.6)

to

Thus, our optimization problem in general can be formulated as follows"

Minimize the functional ff = T + 2 1 ~ i "Jl-2 2 ~ 2 -1-23,~ 3 -}-2 4 ~ 4

(II.10)

over all piecewise continuous functions ai(t),a2(t ) for t >=0 satisfying the inequalities (II.7), where 2, 21, 22, 23, and 2, are certain nonnegati/)e numbers specifying the relative costs of each variable. In the next sections we shall discuss some aspects of the above optimization problem. For simplicity, we shall restrict our presentation to the case f ( t ) - const =fo, whenever 2, = 0. (II. 11)

HI Reformulation of the Dynamical Equations (II.3)* The dynamics of the model is described by Eqs. (II.3)*. For the sake of convenience in our subsequent analysis, we shall reformulate these equations in the following way. Put /)l~a1,/)2~a2

J

9

(III.1)

We now define ~ =(~l, ~v ~3, ~,) T , v=(vp/)2) T , ~ =(0, M - lfo, O, O)-r

(Ill.2) and

0 -M-t(Ke+Kii+Ki2) B;lKil 0

1 -M-iD 0

0 M-1Kil -B-~l(Kil +Kpl )

BE 1Ki2

0

o

M-oKi2

(111.3)

- B 21(Ki2 + Kp2)J

The integral if2 measures the energy cost to the muscles to complete the movement. We assume that the active states aj(t) are piecewise continous and bounded for t > 0 "

and

O0. Finally, if there are perturbing forces which produce Gaussian white noise with variance v on a given trial with p a t h / F , there will be deviations from the path 1I o when v = 0. Let u o = uo(t) describe the path 1I ~ the average response of the system over a number of trials. Then, the net effect produced by the perturbation will be N4 = S [u(t)-- Uo(t)]2dt.

0

= d ~ + N/) + ((~(0) = 0),

tl

~3 = S [nl(t) + nz(t)]dt,

0

(II.9)

where 4 is the (time) derivative of ~ = ~(t). In the case fo = const, the vector ~ is also constant. Let x ~ (x ~ x ~ x ~ x~ -r be a vector such that ~X

0 = ~ 9

(III.6)

The vector x ~ is uniquely determined since ~r is nonsingular. We now put in Eq. (111.5) ~ = X-- X O, X=(XI, X2, X3, X4) T,

(III.7)

which yields

2 = d x + ~/)(x(O) = x~

(Ill.8)

94 Further, by virtue of the constraints (II.7), we have the inequalities f2~ : 0_0) with minimal N4. We assume that xi(T)=h* and x2(T) >=0. We now introduce the additional state variable x 5 = xs(t), the adjoint variables ~k = ~Pk(t)(k = 1, 5), and the Hamiltonian W by the relationships 2 5 = [-X 1 --X0"] 2, X 5 ( 0 ) = 0 ,

(VII.2)

+ q**(T, t, z)v2(z ) + for**(T, t, z)} dz, where T

p*(T, t, z) = ~ P(O- t)p(o - z)do, t T

p**(T, t, z) = ~ q(o - t)p(o - z)dQ, t T

q*(T, t, z)= ~ P(O- t)q(o- z)do, t

r

(VII.9)

q**(T, t, ~)= y q(Q- 0q(o- ~)@, t T

r*(T, t, z) = ~ P(O- t)r(Q- z)d~,

Yf = {B~-t~P3v1 - B~- l~P,v2}

t

+ (Terms which do not involve v 1 and v2), (VII.3)

T

r**(T, t, z)= ~ q(Q - t)r(Q- z)dr

and

t

fPi = M - I(K~ + Kil -t- Ki2)lP2 -- B[ 1Kii~v3 - B 2 1Ki21p4 - 2(x 1 - Xo)~s ~b2= --1/) 1 + M - 1 D l p 2

~P3 -= -- M - aKiltp2 + B I ~(Kia + K p l ) ~ 3

(VII.4)

The functions p(t); q(t), and r(t) are defined by Eqs. (II.19)*. By substituting Eqs. (VII.8) in Eqs. (VII.7) we find a system of integral equations in vl(t) and v2(t) which can be solved numerically. The constant 0 is determined by the equation ~vt~lt=T 0. =

~b4 = -- M - ~Kntp 2 + B~- l(Ki2 + Kp2)~,~

~b5=0

VIII Optimal Control of Oscillations about the End-Point

subjected to the terminal conditions ~vl(T)=0, ~pz(T)=~P3(T)=~P4(T)=0, ~vs(T)= 1, (VII.5) where 0 is a constant. Obviously, we have ~Vs(t) = 1

(0 < t = 0 ) at time t = T

98

by an admissible control vef2~. Let x = x ( t ; v) be the trajectory of the system for 0_< t--- T, and put x 1 = x(T; v), xl(T; v) = h*( =- x~).

(VIII. 1)

Let T be a sufficiently large number such that T>> T, and consider the functional f ~1 = I {[Xl(Z) -- hi'] 2 "4-x2(z)} dz. (VIII.2) T The functional ~1 measures the cost of the oscillations about the end-point for T_< t-< T, and, obviously, we have ~1 =