Biological Cybernetics

Biol. Cybern. 46, 135-147 (1983)

9 Springer-Verlag 1983

Physical Principles for Economies of Skilled Movements* W. L. Nelson Bell Laboratories, Murray Hill, NJ, USA Abstract. This paper presents some elementary prin-

ciples regarding constraints on movements, which may be useful in modeling and interpreting motor control strategies for skilled movements. Movements which are optimum with respect to various objectives, or "costs", are analyzed and compared. The specific costs considered are related to movement time, distance, peak velocity, energy, peak acceleration, and rate of change of acceleration (jerk). The velocity patterns for the various minimum cost movements are compared with each other and with some skilled movement patterns. The concept of performance trade-offs between competing objectives is used to interpret the distance-time relationships observed in skilled movements. Examples of arm movements during violin bowing and jaw movements during speech are used to show how skilled movements are influenced by considerations of physical economy, or "ease", of movement. Minimum-cost solutions for the various costs, which include the effect of frictional forces, are given in Appendices.

1. Introduction

Skilled movements are developed through training and practice to achieve certain objectives associated with the particular task. Finger movements of skilled typists achieve certain speed and accuracy objectives; arm and finger movements of skilled musicians achieve musical objectives associated with such things as tempo, rhythm, tone, and loudness. In the articulation of speech, movements of the jaw, lip, tongue and velum are coordinated to achieve various linguistic objectives. * This paper is based on a talk given at the Engineering Foundation Conference on Modeling in Neuromuscular Systems, Santa Barbara, CA, January 18 22, 1982

In addition to satisfying such primarily taskoriented objectives, many of these skilled movements appear to satisfy a more general, common objective which might be described by such terms as "ease", "economy of effort", or "efficiency". To understand and model such aspects of movements, however, these general terms must be identified with some physical aspects of the movements themselves. How can we identify and quantify the physical parameters and constraints which are relevant to the economy of skilled movements? Obviously, such movements occur in physical timespace coordinates, involve forces and masses, and therefore can, in principle, be modeled as a set of differential equations relating displacement coordinates to the net forces or torques acting along these coordinates. However, even the most detailed such dynamic model is inadequate to explain or predict movements unless it also includes the performance constraints and objectives which affect the neuromuscular control inputs to the model. There are clearly magnitude limits on the forces or torques which can be generated, and on how rapidly they can be changed. The displacements or angles involved in the movement also have physical limits. Constraints on the duration of movements may be explicitly defined (as in musical tempo) or implicitly fixed as a consequence of the force limits. In addition to these constraints on the movements, there is a need to identify the objectives to be attained by the movement. It is possible that some of the objectives may involve constraints; for example, the movement distance may be fixed as a constraint, while the minimization of movement time is the objective. The concept of economy of movement requires that there also be some "costs" associated with the muscular exertion in the movement, the objective being to minimize some measure of cost within the limits of the constraints~

136

It should be emphasized at the outset that there is no intent here to model the complex neuromuscular dynamics, as for example in Hatze (1978). These dynamics are imbedded in the forcing term of our model, upon which only magnitude and rate constraints are imposed. The intent is rather to investigate various performance constraints which may influence the trade-offs being made among competing objectives in the execution of skilled movements.

2. P h y s i c a l C o n s t r a i n t s and O b j e c t i v e s

To illustrate the relationships between physical dynamics, constraints, and performance objectives, a simple model of movements along one dimension will be considered. The formulation is given in terms of linear displacements, but the resulting relationships apply equally well, of course, to angular movements, with torques and moments of inertia replacing forces and masses, respectively. The results can also be extended to those components of more complex movements which can be essentially de-coupled from the other components. The elastic properties (stiffness) of the activated muscles controlling the movement are considered part of the control input to the model. The small reactive forces due to the elastic properties of the inert tissue are considered negligible compared to these active muscle forces (Houk, 1978).

2.1. System Dynamics and Constraints For displacement of a mass m, along the dimension x, with instantaneous velocity, v, the equations governing the motion are

dx -dt

= v,

d(mv) dt

- s

f~(t),

(1)

Jc(t)=v(t), ~(t)=u(t)-bv(t),

x(0) = 0 , v(0)=0,

]u(t)l~ U = Fmax/m,

1 The accelerative and decelerative forces may well have different limit values, since they are generated by different muscle groups. However, since this condition would merely add some complication without significantly affecting the results, the force limits will be assumed equal. Where appropriate, the effect of different force limits will be mentioned

x(7-)=D, v(T)=0,

lu(t)l Tin,

(10)

where V,~ is the minimum V, which also equals the minimum impulse cost for the negligible friction case. The optimal control action, u*(t), corresponds to the slope of this trapezoidal pattern, hence it switches through the sequence U, 0, - U . This form has been described as "Bang-Zero-Bang" (cf., Bryson and Ho, 1969, p. 116). As the movement time, T, is reduced, the minimum impulse pattern changes, but remains trapezoidal in shape, as shown by the dashed-line pattern for T = T' in Fig. 2a. As T approaches T,,, the trapezoidal form converges to the triangular minimum-time pattern indicated by the dotted-lines in Fig. 2a. When friction is included in the movement model (2), the minimumimpulse control pattern can no longer be obtained directly from the velocity pattern constraints. The solution to this optimization problem, given in Appendix B, still has a "Bang-Zero-Bang" form of control. However, the peak velocity is no longer equal to the impulse-cost, I, but is always somewhat greater than it. The instantaneous switching of the control action, u(t), called for in the minimum-cost cases discussed above is, of course, not possible in actual movements. A constraint on the rate of change of acceleration (jerk) is needed to reflect the maximum rate at which muscle force can be changed. The effect of this jerk constraint on the velocity patterns in Figs. 1 and 2 is to cause a "rounding-off' at all points where the slope changes abruptly. 4 2.6. Minimum-Energy Movements The cost measure defined in (5) has been labelled energy cost since there is some evidence (Hatze and Buys, 1977; Hogan, 1982a) that the input power requirement for muscles is proportional to the square of the muscle force output, hence proportional to u2(t). The arbitrary constant, 1/2U, in the energy cost was chosen to provide numerical agreement with the impulse cost, I, under the limit conditions of u(t)= U, O, or - U . The optimal control, u*(t), which minimizes the energy-cost is shown in Appendix C to be a piecewise-linear function of time, in the negligiblefriction case. Depending on the time, distance, and 3 If the accelerativeand decelerativeforce limits are different,the slopes of the limit in Fig. 1 and 2 would also be different.The results in (7), (8),and (10) still hold, however,if U is taken as the product of the two limit values divided by their average 4 Exampleof minimum-impulse velocity patterns with a jerk constraint are given in Sect.2_8 (Fig.4)

139

force limits, the minimum-energy control input may be truncated at the limit values, U or - U. The velocity patterns for minimum-energy control therefore have the quadratic form shown by the solid curve in Fig. 2b, or the composite linear-quadratic forms shown by the dashed and dotted curves in Fig. 2b. Since the movement time, T~, for the dotted curve is only slightly larger than the minimum-time, Tm, this minimumenergy velocity pattern is similar to the triangular pattern shown above it in Fig. 2a. The relationship of the minimum energy-cost, E,~, to the distance, D, movement time, T, and acceleration limit, U, is shown in Appendix C to be

E

1602/UT3' T>=Te , = I U T { 1 {4{1 4D~11/2) IT\-\Sk-UTff/J J' T"

~-" 4O nO h

tt. 20

b=5) :b--o)

w

Fig.

8. Increase in minimum-impulse cost, C=1,~, and the m i n i m u m peak velocity, V, when the frictional damping factor, b, is increased from zero to 5 s - 1 For b = 0, C = V For b = 5, the excess of V over C is indicated by the vertical lines. Acceleration Iimit, U = i9

shown in Fig. 7. Nevertheless, if the movements can be characterized as having an economy of effort as well as time, they should be located above the "knee" region of this surface, where a reasonable trade-off between effort and time is possible. The up-and-down movements of the jaw during speech for a normal-speaking adult should qualify as skilled movements which seem to exhibit a quality of "ease", even when the speech is fairly rapid. On the other hand, it also seems clear that the motor control

of distance and time for each up or down movement is dictated more by linguistic objectives than by any need to minimize some measure of physical effort. It is interesting, therefore, to see if these physical measures have any relevance to such movements. Jaw displacements for two males speaking six English sentences obtained from an x-ray microbeam tracking system (Fujimura et al., 1973 ; Kiritani et al., 1975), were processed to obtain movement times, distances, and peak velocities for both up and down movements of the jaw (Nelson, 1980). The (D, T, V) points for 141 jaw movements of one of the speakers are plotted as the vertical lines in Fig. 7. The vertical line associated with each data point indicates the amount by which the measured peak velocity exceeds the minimum peak velocity bound. The most striking feature of the data in Fig. 7 is the way in which the points are distributed along the "knee" region of the performance surface, which represents the region of good trade-off between movement time, distance, and effort (peak velocity). 7 In terms of these physical measures, the data analyzed so far indicates that there is a consistent economy of movement in the way the jaw moves during speech, even though this is obviously not the primary objective of these movements, a The excess peak velocities, shown by the vertical lines in Fig. 7 are due to at least two factors: (1) the jaw-movement velocity patterns are not generally of the minimum-impulse (trapezoidal) form, and (2) the existence of friction in the movements increases the minimum peak velocity above this ideal (b = 0) lower bound. The inclusion of friction in the movement model also predicts the increasing height of the velocity-excess lines with distance, which is apparent for the data in this figure. Figure 8 shows the theoretical minimum-effort bound and the corresponding minimum peak velocities for minimum-effort movements of various dis~ tances and times when the frictional damping factor in the movement model, (2), is b = 5 s-1, using (B6) and (B7) in Appendix B. The results in Fig. 8 suggest that the excess peak velocities shown in the actual jaw movement data of Fig. 7 are at least partly due to frictional damping effects. 4. Summary and Conclusions In the study of neuromuscular movements, it is important to include the physical constraints and some 7 This result is not sensitive to the choice of U = 1(/. While this value appears appropriate for the jaw data analyzed, the k n e e " region shifts very little when plotted for U =0.5(/or U =2(/ 8 The results for the second speaker were very similar, both in distribution of data points, and in terms of effective acceleration limit

143 measure of the physical objective, or "cost". This paper considered distance and time (duration) as the basic m o v e m e n t objectives, or constraints, and four measures of cost: impulse-cost, energy-cost, peakacceleration cost, and jerk-cost. Of these various costs, the impulse cost was used in the evaluation of jaw movements during speech, simply because of its relationships to a parameter which could be measured with reasonable accuracy - namely, peak velocity. The choice of one particular cost is not crucial to our conclusions, since the minimum-cost b o u n d s based on any of these criteria have a similarly located "knee" region of g o o d trade-off between time and physical cost, and a very steep slope of the minimum-cost b o u n d as time is reduced t o w a r d the minimum-time limit (see Fig. 6). 9 It should be emphasized that the distinction made in Sect. 2 between minimum-time ("Bang-Bang") and minimum-peak-acceleration ("bing-bing") movements is a very important one. The former represent the ultimate exertion type movements which are likely to be reserved for extreme emergency or single, ultimateeffort athletic movements, while the latter represent efficient movements in terms of economizing the peak acceleration (peak force) required to accomplish the movement. It should not be surprising, therefore, to see the triangular shaped velocity patterns of the "bingbing"-type occurring in various repetitive, skilled movements. The actual m o t o r control strategy underlying skilled movements m a y remain hidden from us ; however, we have shown that some of its manifestations in terms of physical performance objectives can be modeled and evaluated. F o r example, the remarkable similarity of movements predicted by the linear-spring model and the minimum-jerk model (see Fig. 3), plus the fact that the energy-cost of b o t h such movements is less than 2 % higher than the minimum-energy-cost (11) p r o m p t s an interesting speculation: might the m o t o r - c o n t r o l p r o g r a m m i n g of net muscular forces in repetitive, skilled movements be such as to produce a nearlyconstant stiffness condition (in spite of the nonlinearities in the force/length characteristics of muscles) because of the physical benefits of achieving movements which are both optimally s m o o t h and energyefficient? A study to test this hypothesis on the jaw m o v e m e n t data described in Sect. 3, is currently being conducted by the author.

The evaluation of m o v e m e n t in terms of competing objectives, such as distance, time, and some measure of m o v e m e n t cost, can be very useful in understanding the way in which these objectives are balanced, or traded-off, in the execution of skilled movements. The relationships between m o v e m e n t distance and time for skilled movements m a y be governed by m o t o r control strategies designed to fulfill the primary objectives of the task independent of considerations of physical economy. However, we have observed that there is at least some a c c o m m o d a t i o n to physical e c o n o m y in skilled movements. F o r example, the jaw m o v e m e n t data discussed in Sect. 3 has illustrated that the m o t o r control strategy is likely to be influenced in a very sensible way by physical objectives which relate to the movement's effort or energy. Further, it might be that the primary objectives themselves have evolved in conformity with such physical objectives. Acknowledgements. The author is grateful to Joan Miller, Belt Laboratories, and Paul Zukofsky, Musical Observations, Inc. for the use of the arm movement data on violin bowing. Acknowledgement is due to the staff members of the Research Institute of Logopedics and Phoniatrics, University of Tokyo, and to Osama Fujimura, Bell Laboratories, for providing the X-ray microbeam data from which the jaw movement data was obtained.

Appendices A. Optimal Control Methods and Minimum-Tune Solutions

The movement time, T, and the distance, D, are the principal measures of performance for movements governed by (2) in Sect. 2 of the paper. A general measure of movement cost of the types given in (4)-(6) can be defined as T

0

where c(x, v, u) is some scalar function of the instantaneous values of displacement, velocity, and control action. If a control action u*(t), is found which, in addition to satisfying the physical system constraints, also minimizes the cost, C, it is said to be an optimal control. For the system described Sect. 2, the modified form of the Hamiltonian, as defined by Pontryagin et al. (1962), is 1~: H(x, v , p , q , u ) = p k + @ - c ( x , v , u ) = p v - qbv + q u - c(x, v, u),

(A2)

where the relations (2) have been substituted to obtain the second form of (A.2). The adjoint, or costate, variables p and q are governed by p = - 8H/Ox,

9 If reliable estimates of peak-acceleration, energy, and "jerk" costs (as defined in Sect. 2) could be obtained from the measurement of various types of movements, they would certainly be of interest in evaluating the extent to which they are being optimized in the process of controlling skilled movements. However, for energy and jerk costs, there is a possibility of large errors introduced by the squaring of successive differentiations of displacement data

(A1)

C = ~ c(x, v,u)dt,

~1= - ell~By,

(A3)

where the initial or terminal conditions on (p, q) are unspecified, since (x, v) have fixed initial and terminal conditions. The maximum principle states that, for a set of time functions (2, ~,/3,q, fi) which 10 Many formulations, such as Athans (1966), use +c(x,v,u) instead of - c ( x , v, u) in which case H is minimized rather than maximized, yielding identical results

144 satisfy (A2) and (A3), if ~ is an optimal control, in the sense of minimizing the cost, C, then for each ts[0, T],

u=-U

for S(6D/U)~/2"

(C4)

The peak velocity occurs at t = T/2, and has the value,

V=3D/2T,

T>=(6D/U) 1/2 ,

(C5)

i.e., the peak velocity exceeds the average velocity by 50%. The minimum-energy cost, Era, for this case is, from (5) and (C4),

E,,=6D2/UT 3 ,

T>(6D/U) 1/2 .

(C6)

which is equivalent to (10) in Sect. 2.5. When the friction term in the equatmns of motion, (2), is negligible, the acceleration, a(t), is equal to the applied control input u(t). In this case, the impulse-cost in (4) can be written as

For the movement times less than (6D/U) 1/2, the control function is saturated over the intervals (0,R) and ( T - R , T), where R is the instant at which q(t) = 1. The Eq. (2) can be integrated in parts, using the minimum-energy control law (C3). The results of this integration and matching of boundary conditions is that the minimum-energy cost, E,~, can be evaluated from (5) as

1r I = ~ ! If(t)ldt.

UT { /4 / 4D E m= ~ - I 1 - l~II-~TS))

T = D/V,, + V,,/U

(BS)

From (2), the velocity is zero at t = 0 and t = T. If, in addition, the velocity is unimodal, i.e., its slope has only one reversal of sign in the interval (0, T), then (B8) can be written in the form

where R is the instant at which the velocity slope, ~, changes sign or becomes zero. Since this must also be the instant at which the peak velocity, V,, is attained, Eq. (B9) reduces to 1/,,(m

I=~[ ! dr-

~(T)

\

1

y dv)= ~(V + V)= V, v(R) /

(BIO)

Hence, for all movements satisfying the conditions of negligible friction and unimodal velocity patterns, the peak velocity is numerically equal to the impulse-cost of the movement. All departures from these conditions require peak velocities somewhat greater than the impulse-cost.

C. Minimum Energy Movements Setting c(x, v, u) = uZ/2U in Eq. (A1) of Appendix A, gives the energycost as defined by (5) in Sect. 2.2 of the paper. For the equations of motion given in (2) of Sect. 2.1, the energy-cost version of the Hamiltonian in (A2) becomes,

H(x, v, p, q, u) = p v - qbv + qu - u2/2 U

(C 1)

with the constraint, lu(t)l _-