J. Biompchlmrcs.

1977. Vol.

IO. pp. 799405.

Pergamon

Press.

Pnnted

III Great

Britain

A COMPLETE SET OF CONTROL EQUATIONS FOR THE HUMAN MUSCULO-SKELETAL SYSTEM* HERBERT HATZE National Research Institute for Mathematical Sciences, CSIR Pretoria, South Africa Abstract-A mathematical model of the total human musculo-skeletal system is presented. The model comprises a link-mechanical and a musculo-mechanical set of ordinary first-order differential equations which deacrihe the dynamics of the segment model and muscle model respectively. The interdependence

of the two sets of equations is demonstrated. The set of musculo-mechanical quations contains the two neuromuscular control parameters motor unit recruitment and stimulation rate, and the significance of such a representation for a control-theoretical treatment of musculo-skeletal systems is discussed. Finally, after a short discussion of the successful application of the present model in the prediction of an optimal human motion, further possibilities are indicated of the use of the model for investigations into the control behaviour of musculo-skeletal systems.

INTKODUCTJON There appears to be increasing interest in optimal processes of the human and animal musculo-skeletal

system. Some of the more recent attempts to predict optimal transition processes for such systems have been reasonably successful (Chow and Jacobson, 1971 and 1972; Hatze, 1976) while others were less effective in obtaining optimal solutions (Zomlefer, Ho, Levine and Zajac III, 1975; Ghosh and Boykin, 1976). The approach usually adopted is to derive the mechanical equations of motion of the biosystem under consideration, and to identify the muscle torques with the control parameters of the system. The intricate and highly nonlinear internal excitation and contraction dynamics of the muscle is frequently completely ignored. It is, therefore, not very surprising that the majority of these models fail to produce predictions of optimal processes. This paper is &voted to the derivation of the complete set of differential equations which describe the dynamic behaviour of the total human musculoskeletal system, including the excitation and contraction dynamics of each of the muscle groups acting on the system. Moreover, the equations will also contain the two neuromuscular control parameters motor unit recruitment and stimulation rate, for each muscle involved. It will be possible to distinguish between a link-mechanical part and a musculo-mechanical part of the complete system of equations of motion. Functionally, the two sets are, of course, interdependent. THE SET OF LINK-MECHANICAL EQUATIONS OF MOTION

Mechanical models of the segmented human body have appeared frequently in the literature (Fischer, 1906; Krogman and Johnston, 1963; Nubar and Contini, 1961; Beckett and Chang, 1968; Kane and Scher, * Received 21 December 1976.

1970; von Gierke, 1971; Seireg and Arvikar, 1973; Hatze, 1973. 1976; Chao and Rim. 1973 : and others). The differential equations describing the dynamics of the model are usually obtained by applying Newtonian mechanics (or d’Alembert’s principle) or the Lagrangian formalism in more sophisticated treatments. The latter formalism will be used to derive the set of link-mechanical equations of motion for the present model. When an attempt is made to define a model of the human skeletal link system, it is soon realized that considerable simplifications have to be adopted if the procedure is to lead to feasible results. Kinzel, Hall and Hillberry (1972) have demonstrated that in each human joint a total of six degrees of freedom is permitted to some extent. They remark that the accuracy with which the motion is described will deteriorate as the number of degrees of freedom considered is decreased. This is certainly so but one should keep in mind that there are some 144 joints in the body, if the minor ones are included. Clearly, if all these joints, each of which is considered to exhibit six degrees of freedom were included in the model structure, this would lead to a chaotically complicated system. Hence simplifications have to be introduced, the question being only how many, where and under which circumstances. Generally speaking, if the model is to be used for the simulation of gross motions only, only major joints of the body with a maximum of three degrees of freedom (depending on the type) need to be considered For more delicate motions (e.g. writing) the appropriate refinements have to be made. These remarks, however, hold true only for the skeletal-link part of the system. When the muscle leverage is under discussion, all six degrees of freedom of the joint in question must usually be taken into account (see below). If only gross body motions are to be considered, a division of the body into 17 segments may be regarded as both reasonably accurate and yet not too complicated. The segments are considered as rigid

799

HERBERT HATZE

800

bodies connected by joints with varying numbers of degrees of freedom. This segmented model of the human body is similar to the one proposed by Hanavan (1964) and used by Huston and Passerello (1971) and Huston, Passerello, Hessel and Harlow (1976) except that it contains the shoulders as separate segments. In addition, the present model places no restrictions (apart from certain symmetry requirements) on the morphology of the segments. This is in contrast to Hanavan’s model where the segments are elliptic cylinders, spheres, cones or truncated elliptical cones. The model, together with the inertial reference frame OXYZ, is depicted in Fig. 1. We shall now derive the link-mechanical equations of motion of the biosystem. It will be shown that this can be done in a less tedious and more elegant way than that usually adopted in biomechanics literature. Let the position of the reference point 2 = (X, I: 2) of the body model with respect to the inertial reference frame OXYZ be defined as the midpoint of the horizontal cross-section through the body at the height of the navel and let the orientation of each segment relative to the proximal segment be given by (at most) three Eulerian angles x, a = fl, /I + 1, fi + 2 (fl some integer). In addition, let the superscripts E and 6 refer to positions as follows: l denotes left (E = l), middle (C = 2) or right (E = 3), and 6 upper (8 = 1) and lower (6 = 2), respectively. Then any position vector pj’,6’ of the centre of mass of the i(e, d)-th segment can be expressed as the sum of the reference vector & and the appropriately transformed internal position vectors of the distal joint centres and centre of mass in the case of the last vector respectively. That is,

where ,J@*~)(xJ denotes a transformation matrix (in terms of Euierian angles x,) of the form given on

p. 345 of Wells (1967). and $*d’, j = 1, . . . , i(c, 6) - 1, are the internal position vectors (with components along segment-fixed axes) of the distal joint centres. The vector @d& is the internal position vector of the centre of mass of the i(c, 6)-th segment. Note that double-indexing (c. 6) of the index i was essential in order to distinguish between the left, middle and right, and upper and lower parts of the model. This is a consequence of the special choice of the reference point in the body, which choice, in turn, was made with a view to minimizing the number of transformation matrices Af*” (x,) to be multiplied. As regards the angular velocity vector @pd) of the i(e, 6)-th segment, it must be remembered that this vector is measured relative to the inertial frame OXYZ, while its components must be expressed relative to segment-fixed axes. Thus we have &.d’

=

&vd)

+

1 i’r*d)-l

[ fi

j=l

al;.~~;_l+,(x,,]

I=1 x

f&;-p

(2)

where &*d) denotes the spatial angular velocity vector of the i(e, S)-th segment and A(‘,d)r is the transpose

of the transformation matrix A(‘*@ appearing in expression (1). The vector ,icr*a) mu? be expressed in terms of the relevant Eulerian angles and their time derivatives (see Wells, 1967, p. 157). It is now easy to obtain the Lagrangian function Lfor the model under discussion. Let it be assumed that all body segments possess an axis of symmetry (which does not imply that the segments have to be isomorphic). Then all three segment axes are principal axes and if the origin of the system of segment axes lies on one of the principal axes passing through the centre of mass of the segment, all products of inertia are zero. Let pm denote the inertia tensor of the i(c, 6)-th segment with respect to axes which pat% through the centre of mass. In our special case this tensor reduces to the diadic form i!f. 61= i,r.“) i.i + i,f.d) i.j + __ -I

ipk.k,

(3)

where i, i and k are unit vectors along the respective segment axes. Let, furthermore, MrSd) and g denote the mass of the i(e, 6)-th segment and the gravitational constant, respectively. Then the kinetic energy of the i(c, 6)-th segment is given by ??‘.d’

I

=

~~(i(,d)~~.d).~~.d)

+

(4)

&$.d’.f~.d).&d),

and its potential energy by v!f.d’

I

Fig. 1. Segmented model of the human body. The numbers next to the joints indicate the respective degrees of freedom. An inertial reference frame is defined by the orthogonal system of axes OX, OY and OZ. The vector (X, x Z) defines the reference point of the body model while the vector

pi denotes

the inertial space position .-. . .

centre 01 mass 01 tne I-m segment.

of the

=

Ml’.d’

g k. &cd)+

(5)

The Lagrangian L for the total link-mechanical system is therefore

_

gk.

&.d’)

+

&,l’.”

. If.“‘.

&.d’,

(6)

Control equations for the human musculo-skeletal system

where

(23)

1j(-uz4) = ~j(x*)b~j(X(x-,)]1'2,

(24)

and

The transpose AT(xJ of the transformation matrix ,4(x,) is the same as that defined in expression (2). The force vector Er can now be written as

njl;ii =

where the magnitude Fy of the muscle force can obviously (see Fig. 2) be expressed as the sum of the force Fjp”[(l,&). /J.s,.s;)] across the parallel-elastic element and the force fy (I,&), ,Ij) across the series elastic element. i.e. ) 1 ,), (26) , J “‘I Fy = F,PE(lf.Y,).ip,. ,x.1)) + FsE(I(.~

l,....m,

)3 X2n+m+j(")=

x%+m+j9 X2m+2m+j

C2jX2n+3m+j

i2n+3m+j=

C2j(C3jvj-X2n+m+j)Erl(X2n+j)~~j(~

(C3jVj-X2n+m+j)>

(")=xlR+2m+j9

-qOj)+

40jwX2n+2m+j)

-

r*(X2m+j)X2n+3m+j],

(32)

-Y2n+3m+i(o)=.Yqn+3,+j.

The torque produced by the j-th muscle, and taken relative to n&c, is easily seen to be (see Fig. 2) #

= Bj - -p(\-,)I x 1,“. (27) which by virtue of equations (22)-(27) takes the form a,) + Fy(x,, Aj)][(crj - @.xX)) x /P(S’)(Sj - C\(S,))]lj(S,). (28) -

In general. the number of muscles, say w, situated across a joint. is greater than one. The total torque vector due to muscular action is then NM = f

IY’r,

(29)

j=l

with components NY, N,” and NY, respectively. Note that these components are not identical with the generalized forces QFXof this joint because the latter are directed along the respective angular velocity vectors of the Eulerian angles and these vectors do not coincide with the axes nt&. However, the necessary transformations can easily be performed and by putting x, = JI, 0, 4 we find (Wells, 1967, p. 158) that

I[

Q? # [

Qf

=

sin@sin$

sinBcos4

cos6

cos4

-sin4

0

0

1

0

0

S2n+,.

4X 2n+j Ir 2n+2m+j

X2n+m+j

X2n+2m+j=

j$’ = [F+,.

(31)

where 5 is a muscle-specific constant, and let xzn+,,,+j, xzn+zm+j and xzn+3,,,+jrj = 1,. . . . m, denote three variables which describe the excitation dynamics of the j-th muscle (for details see Hatze, 1977a). Then it can be shown that the internal dynamics of the j-th mUSde can be completely described by the drfferential system (for j = 1,. . _, m) XZn+j(O) =

alj

(

c2j c3jvj-

j=

[F~(x,,x~~+~)/F~ + b,je-ub,(\‘“*l-I)]bzj _ 1

1

iln+j = C,j a2j + -arctanh X2n+m+j=

X2n+jr

where ali, azj. ahj, b,+ b,j, clj, c2ja c3i. Fj. and qoj are muscle-specific constants; k(xln+J_ rl(xZn+& and r2(xz.+3 are the length-tension and active-state length dependence functions respectively: and Uj and Uj are the control parameters of the j-th muscle denoting the relative number of active motor units and the relative average stimulation rate respectively (for exact definitions see Hatze, 1977a).The control constraints are given by 0 < zri.r*iG 1, j = I,.... m. (33) and thus define a 2m-dimensional parallelepipcd. The first of equations (32) describes the contraction dynamics of the j-th muscle while the remaining three describe its excitation dynamics. The complete set of 4m differential equations which represent the dynamic behaviour of all the m muscles in the total model, and which contain the 2m neuromuscular control parameters 11~.17~.,j = I . .. . . III, constitute the scr qf musculo-mechanical equations oj’ motion.

NY , I[I NY

(30)

NY

which establishes the final relationship between the generalized muscle torques Qx”,V_,of equations (10)

It should perhaps be mentioned that equations (32) incorporate such well-known functions as the forcevelocity relation for the shortening as well as lengthening of the contractile element. the length-tension relation, the dynamics of the active state, the nonlinear dependence of the force production on the stimulation rate, and all the complicated interrelationships which exist between the excitation and contraction variables.

804

HERBERT HATZE

There is, however. a limitation imposed on the allowable dynamics of the controls up Although these controls can assume any value between zero and unity, this value must remain constant during a simulation run. This is a consequence of the conditions under which the model was derived (see Hatze, 1977a). However. this restriction does not apply to the controls rj. nor does it apply to the simplified version of the muscle model (Hatze, 1977a) which has been used successfully in the optimization of a human motion (see below). Efforts are at present being made to extend the exact model to general control states.* On integrating the system (32) on an IBM 370/158 digital computer certain numerical stability problems were encountered upon switching the controls uj discontinuously from unity to 0 (bang-bang controls). After some investigations it turned out that these problems can be overcome by applying the scaling transformation X2n*m+jC IObXzn+mj3 C_3j+ 106CJj, 10-12rl(x2n+j)r

rl(X2n+j)c

X2n+3m+jC10-6X2n+3m+j~ ad

dx2.+j)

+

10-6r2(~2n+jh

ad

by

using the integration subroutine DVOGER from the IMSL-package with an error bound of 1 x 10-s and with MTH = 1. The connection between the link-mechanical and the musculo-mechanical part of the model is provided by the function (for derivation see Hatze, 1977a) Fy(.v,, x2.+ j) = co,(eCIi(C6j’~X.)-n*“*,+C,i) - 1). (34) in which cy - C7j are constants. This function (34) appears in both the system (32) and the system (10) via the functions (28), (29), and (30). In fact, the appearance of FTE(x, xlR+j) in (32) is the result of the substitution T(.f

2n+j*X2n+j9X2n+2n+j)=

FjSE(XZTX2n+j).

(35)

which holds true by virtue of the arrangement shown in Fig. 2. It is thus seen that at each moment of time the state of all the muscles in the system is influenced (via the variables x, L-Y < 2n) by the state of the linkmechanical system part while, conversely, the state of all the muscles in the system determines the behaviour (via the variables x2”+> j = 1,. . . , m) of the linkmechanical part. This state of affairs is clearly seen when the complete set of(2n + 4m) control equations for the total musculo-skeletal system is written down in abbreviated form, thereby combining the systems (7). (lo), and (32):

* Such a general myocybernetic control model has now been developed (see Hatze, to be published). It removes the constraints on the allowable dynamics of the controls u,. and also eliminates the stiffness problems encountered when numerically integrating the present system (32) It is recommended that for practical applications the excitation dynamics in (32) be replaced by the excitation dynamics of the new general model (available from the author).

X2,i2r

1 =

r =

Xtrr

= f2AXm

x2n+

l,...,n, a,r=l,....

jb

n;j=l,..., a =

~2n+j=fZn+j(X=.X2n+j.X2n+2n+jb

j=l X2n+m+j -x2n+2m+j-

l,...,n,

. . . . . m, f 2n+m+j

= -1‘

2n+Zm+j

.j= i2n+3m+j

m.

2n+m+p ‘;_

1,...,

=f2n+3m+j X2n+3m+j9

j =

1, . . . . m. “j).

-~2n+m+j.S2n+3m+j*

m. fx2n+* Cj. Ujh

X2n+m+jt

j =

l,....

X2n+2m+b

m.

(36)

The first two of equations (36) comprise the linkpart while the last four constitute the musculo-mechanical part of the total system.

mechanical

DkSCUSSlON AND CONCLUSION It is believed that the model presented here is the first mathematical model of the total human musculoskeletal system which contains as control parameters the two independent controls actually employed by the nervous system to grade the force output of the muscles: motor unit recruitment and stimulation rate coding. The model has already passed its most severe test: a slightly simplified version was successfully used in the prediction of a time-optimal motion of a subsystem (consisting of the right leg) of the total human musculo-skeletal system. In this project (Hatze, 1976) optimal control theory and a first order algorithm of differential dynamic programming (Jacobson, 1968) were used to predict the optimal controls as well as the optimal trajectory of the model of the biosystem, subject to certain boundary conditions. The optimal model solution was then compared with the performance of the living system and it was found that any motion of the biosystem which deviated from the predicted optimal one took a longer time to complete and hence was not time-optimal. In addition, the trajectories and control functions of near-optimal motions as measured on the living system were indeed found to be very close to the theoretical optimal process, thus again confirming the optimality of the model solution. One of the advantages of the present model is that the controls are directly related to observables (electromyograms) of the biosystem. This fact greatly facilitates any comparison between the model predictions and the processes as measured on the living system. Finally, the model is expected to contribute substantially to our understanding of the control behaviour of musculo-skeletal systems. Indeed, a necessary condition for optimality of motions involving ‘maximum effort’ (which is defined in terms of a certain set of performance criteria) is that the controls u, be of bang-bang form. Recent experimental results (Hatze and Hayes, to be published) seem to confirm this prediction although only approximations to the actual bang-bang controls could be observed. As far as

Control equations for the human musculo-skeletal system energy minimization is concerned. Hatzc and Bu>s (1977) have recently succeeded to show that the

specific pattern of rate coding and motor unit recruitment observed in isometric contractions in man (Milner-Brown. Stein and Yemm, 1973) may be attributed to a minimum-energy criterion. In addition, the size principle of motor unit recruitment has been demonstrated (Hatze, 1977b) to be the realization, by the neuromuscular system, of a general teleological principle of optimal force grading sensitivity.

REFERENCES

Beckett, R. and Chang, K. (1968) An evaluation of the kinematics of gait by minimum energy. J. Biomechanics 1, 147-159. Chao, E. Y. and Rim, K. (1973) Application of optimization principles in determining the applied moments in human leg joints during gait. J. Biomechanics 6,497-510. Chow, C. K. and Jacobson, D. H. (1971) Studies of human locomotion via optimal programming. Moth. Biosci. 10,

x05

Krogman. W. M. and Johnston, F. E. (1963) Human mechanics. Four monographs abridged. Techn. Dot. Rept. No. AMRL-TDR-63-123, Wright-Patterson Air Force Base. Ohio. Milner-Brown. H. S.. Stein. R. B. and Yemm. R. (1973) The orderly recruitment of human motor units during voluntary isometric contractions. J. Physioi. 230. 359-370.

Nubar. Y. and Contini. R. (1961) A minimal principle in biomechanics. Bull. Math..Biopkys. 23. 337-391. Seireg, A. and Arvikar. R. J. (1973) A mathematical model for evaluation of forces in lower extremities of the musculo-skeletal system. J. Biomechics 6. 313-326. Von Gierke. H. E. (1971) Biodynamic models and their applications. J. Acoust. Sot. Am. SO, 1397-1413. Wells, D. A. (1967) Theory and Problems oj’ Lagrangian Dynamics. p. 345. Schaum Publ. Co., New York. Zomlefer, M. R., Ho, R.. Levine, W. S. and Zajac III. F. E. (1975) A studv of the coordination of cat hindlimb muscles during a maximal vertical jump. Proc. IEEE Co& on Decision and Control. Houston. TX. pp. 402-407.

239-306.

Chow, C. K. and Jacobson, D. H. (1972) Further studies of human locomotion: Postural stability and control. Moth. Biosci. 15, 93108. Fischer, 0. (1906) Theoretische Grundlagen fir eine Mechanik der lebenden Kkper, pp. 52-55. Teubner, Berlin. Ghosh, T. K. and Boykin, W. H. (1976) Analytic determination of an optimal human motion. J. opt. Theory Appl. 19, 327-346. Hanavan, E. P. (1964) A mathematical model of the human body. Aerospace Med. Res. Lab. Report No. AMRL-TR-64-102. Hatze, H. (1973) Optimization of human motions. In Biomechanics lfl (Edited by Cerquiglini S.. Venerando. A. and J. Wartenweiler), pp. 138-142. Karger. Base]. Hatze, H. (1976) The complete optimization of a human motion. Mom. Biosci. 28. 99- 135. Hatze. H. (1977a) A myocybernetic control model of skeletal muscle. Biol. Cpbern. 25. 103-l 19. Hatze, H. (1977b) A teleological explanation of Weber’s law and the motor unit size law. CSIR Special Rep. WISK 248. Hatze, H. (to be published) A general myocybernetic control model of skeletal muscle. Bin/. Cyherrl. Hatze. H. and Buvs. J. D. (1977) Energy-optimal controls in the mammalian neuromuscular system. Biol. C_vberrl. 21. 9-20. Hatze, H. and Hayes. K. C. (to be published) Approximation of bang-bang controls by the human neuromuscular system (in preparation). Huston. R. L. and Passerello. C. E. (1971) On the dynamics of a human bodv model. J. Biomechanics 4. 369-378. Huston, R. L., Passkrello, C. E., Hessel, R. E. and Harlow, M. W. (1976) On human body dynamics. Ann. biomed. Engng 4, 25-43. Jacobson, D. H. (1968) New second-order and first-order algorithms for determining optimal control: a differential dynamic programming approach. J. opt. Theory Apppl.. 2, 411-440. Kane, T. R. and Scher, M. P. (1970) Human self-rotation by means of limb movements. J. Biomechanics 3, 39-49. Kinzel, G. L., Hall, A. S. and Hillberry, B. M. (1972) Measurement of the total motion between two body segments-1. Analytical development. J. Biomechanics 5, 93-105.

NOMENCLATURE

external position vectors of the reference point of the body model and the centre of mass of the i(c. 6)-th segment respectively. !t~d’ internal position vectors, with components along segment-fixed axes, of the distal joint centres of the j(e, b)-th segment. If j = i(e, S) then bp” denotes the position of the centre of mass of the j(r. 6)-th segment l,d superscripts indicating the position. relative to the reference point, of the segment in question s,. r = I..... 211+ 4~1. state variables of the complete model of the musculo-skeletal system n, m number of link-mechanical degrees of freedom and number of muscles present in the system respectively Aj’.6’ (x,) transformation matrix. in terms of Eulerian angles .Y,. from the coordinate system of the I(r. &th to the coordinate system of the [/(E. &I]-st segment c&“’ angular velocity vector of the i(e. 6)-th segment f(f.d’ inertia tensor of the f(e, 6)-th segment with respect -I to axes passing through the centre of mass i, j, Ir unit vectors along the segment axes Ox. Or and 0.2 respectively My.“’ mass of the i(e, S)-th segment g gravitational constant (g = 9.81 msec-‘) ‘1I’-“‘. l’)r.“’ kinetic and potential energy of the i(E. bbth segment respectively Qx_ generalized force corresponding to the coordinate x, zj, rr, internal position vectors of the origin and insertion of the j-th muscle respectively q(x,), e(x.) internal position vectors of the non-stationary centre of rotation C @ force vector produced by the .j-th muscle CEj, SEjg PE, contractile element, series-elastic element and parallel-elastic element respectively, of the model of the j-th muscle ;.,.;.,,.I, lengths of the j-th contractile. series-elastic and parallel-ekistic elements respectively FsE I ’ FTE, Fy forces across the respective elements of the j-th muscle $, 0, #J three specific Eulerian a_ngles ai,, ash ash b,,, b,j, Clj,. , C,jr Fj. qoj specific constants of the j-th muscle u,, uj neuromuscular control parameters of the j-th L pi “*”

muscle.