Biol. Cybern. 76, 321—329 (1997)

Invariant characteristics of horizontal-plane minimum-torque-change movements with one mechanical degree of freedom Sascha E. Engelbrecht 1, Juan Pablo Ferna´ ndez 2 1 Department of Psychology, University of Massachusetts, Amherst, MA 01003, USA 2 Department of Physics and Astronomy, University of Massachusetts, Amherst, MA 01003, USA Received: 3 September 1996 / Accepted in revised form: 7 January 1997

Abstract. Uno et al. (1989) suggested that movements are organized such that the squared change of torque is minimized over time. Although influential, this theory has attracted much less attention from experimental researchers than the competing minimum-jerk model (Flash and Hogan 1985). One reason for this relative neglect has been the lack of general quantitative predictions, which results from the belief that minimum-torquechange trajectories have to be computed numerically for individual movements and arm-dynamical parameters. In the present paper, we show that for an important special case, that of planar horizontal movements with one mechanical degree of freedom (DOF), it is actually possible to find an analytic expression for the predicted minimum-torque-change trajectories. Based on this mathematical result, we derive a set of properties which are characteristic of these trajectories and compare them to experimental data which have not previously been related to the minimum-torque-change model. Certain discrepancies between these experimental data and minimum-torque-change model predictions are revealed.

1 Introduction Most motor control tasks are mathematically ill-defined; they allow an infinitely large variety of responses compatible with the task requirements. Consider, for instance, the task of reaching for a point target. First, there is an infinite number of geometric curves (paths) along which the hand could be moved towards the target. Second, there is an infinite number of ways of timing the progress of the hand along any of these curves. Third, there is generally an infinite number of joint configurations corresponding to each spatial location of the hand. Correspondence to: S. E. Engelbrecht, Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA (e-mail: [email protected])

Although point-to-point reaching tasks are extremely underconstrained, human subjects show little within- and between-subject variability when performing such tasks. By the mid-1980s, a wealth of research had accumulated indicating that hand paths are approximately straight (Morasso 1981, 1983; Flash and Hogan 1985) and that hand tangential-velocity profiles are single-peaked, bellshaped, and essentially symmetrical (Atkeson and Hollerbach 1985; Flash and Hogan 1985; Morasso 1981). Based on the work of Flash (1983), Hogan (1984), and Nelson (1983), Flash and Hogan (1985) summarized these findings in a powerful theory which suggested that hand trajectories are maximally smooth or minimally jerky ( jerk is the time derivative of acceleration). For point-topoint movements, this resulted in the predictions that hand trajectories (1) follow straight lines, (2) scale linearly with movement amplitude, (3) are invariant under translation and rotation of the movement endpoints, (4) are shape-invariant under changes in movement duration, and (5) have symmetrical, single-peaked, smooth velocity profiles. Since these predictions were consistent with the majority of the experimental data available at the time, the minimum-jerk theory became quite influential in the motor control literature and was popularized by Flash and Hogan in numerous publications (Flash 1990; Hogan 1988; Hogan and Flash 1987; Hogan and Winters 1990). Various later findings, however, revealed discrepancies between minimum-jerk trajectories and observed behavior. In particular, there was increasing evidence that hand paths are curved for certain point-to-point reaching tasks and that hand-path curvature systematically varies with workspace location (Atkeson and Hollerbach 1985; Flanagan and Ostry 1990; Flash 1987; Kaminsky and Gentile 1986; Lacquaniti et al. 1986; Uno et al. 1989). Motivated by the minimum-jerk model’s failure to explain these workspace variations in hand-path curvature, Uno et al. (1989) proposed the following alternative optimality principle: Movements are organized such that the time integral of the squared sum of torque changes is

322

minimal. The minimum-torque-change principle predicted the observed variations in hand-path curvature. Philosophically, it represented a drastic break with the classic hierarchical view (Saltzman 1979) in which movement planning (hand and joint kinematics) was independent of and superordinate to movement control (torque generation), and its proposition initiated an extensive and continuing controversy between the proponents of the minimum-jerk and the minimum-torque-change principles (see Kawato 1996 for a review). This controversy almost exclusively dealt with the issue of path curvature. Advocates of the minimum-jerk model attempted to establish that the geometric shape of hand paths was a function of perceived curvature (Wolpert et al. 1994) but not of arm dynamics (Flash and Gurevich 1991; Shadmehr and Mussa-Ivaldi 1994). Supporters of the minimum-torque-change model tried to show the opposite: Hand paths should be independent of perception (Uno et al. 1994) but dependent on arm dynamics (Uno et al. 1989). The controversy did not address potentially different predictions of the minimum-jerk and minimum-torquechange models concerning the temporal aspects of movement. In fact, Uno et al. (1989) stated that minimum-torque-change trajectories and minimum-jerk trajectories are quite similar in their temporal characteristics, and this view was reiterated by Kawato (1996). It is, however, not known if these similarities persist under variations in movement duration or under changes in arm dynamics. This uncertainty is partly the result of an unfortunate feature of the minimum-torque-change principle: There is no analytical solution to the problem of finding the trajectory which minimizes torque change for multijointed arms. Particular solutions may be found via extensive numerical computations (Uno et al. 1989), and much effort has been expended on devising neural mechanisms which could plausibly carry them out (Kawato 1992; Kawato et al. 1990; Wada and Kawato 1993). The numerical predictions, however, lack generality because they are necessarily restricted to a particular movement (defined by an initial arm posture and target location) and to the researchers’ estimates of dynamic parameters (inertias, viscosities, and segment lengths). This lack of general quantitative predictions may be responsible for the fact that the many experimental investigations of the minimum-jerk model (Furuna and Nagasaki 1993; Hreljac and Martin 1993; Nagasaki 1989, 1991; Schneider and Zernicke 1989; Viviani and Flash 1995; Wann et al. 1988; Wiegner and Wierzbicka 1992) have not been matched by a comparable amount of experimental work on the minimum-torque-change model. Although it is impossible to find an analytic expression for minimum-torque-change trajectories in the general case of unrestricted arm movements — where seven mechanical DOF are involved — such a solution does exist for one-DOF planar horizontal forearm movements. A considerable body of experimental data about these is available (Baba and Marteniuk 1983; Cooke and Diggles 1984; Cooke et al. 1989; Darling and Cooke 1987; Flament et al. 1984; Moore and Marteniuk 1986;

Nagasaki 1989, 1991; Shapiro and Walter 1986; Sherwood et al. 1988; Wiegner and Wierzbicka 1992). In the present context these movements are worth considering because the minimum-torque-change theory applies equally to all arm movements, regardless of the number of mechanical DOF involved. One DOF planar horizontal movements are thus as relevant in the context of this theory as are the two-DOF planar horizontal movements studied in Uno et al.’s (1989) original work.

2 An analytical solution of the minimum-torque-change criterion for planar horizontal movements with one degree of freedom The dynamics of one-DOF horizontal-plane arm movements are described by the ordinary second-order linear differential equation N"Ih® #BhQ

(1)

where N denotes torque, I is the inertia of the limb segment, and B is the viscosity coefficient, assumed to be independent of the state of the arm. Equation (1) is a restatement of Uno et al.’s (1989) arm model for the case of one-DOF movements. From (1), it follows that the squared rate of change of torque is NQ 2"(Ih® Q#Bh® )2"I2h® 0 2#B2h® 2#2BIh® h® 0

(2)

The minimum-torque-change principle suggests that movements are organized such that the time integral of the squared rate of change of torque t

2

C": NQ 2dt t

(3)

1

is minimal. For this to occur, the Euler-Lagrange equation d3 LNQ 2 d2 LNQ 2 ! "0 dt3 Lh® Q dt2 Lh®

(4)

must be satisfied. Insertion of (2) into (4) yields the ordinary sixth-order linear differential equation h(6)(t)!b2h(4)(t)"0,

b,B/I

(5)

which can be reduced to one of order two via the substitution x"h(4). The solution of (5) can therefore be written as h(4)(t)"x"u b4 sinhbt#u b4 coshbt (6) 1 2 and the angular-position curve is obtained by quadruple integration: h(t)"u sinhbt#u cosh bt#u t3 1 2 3 #u t2#u t#u 4 5 6

(7)

323

Equation (7) has six unknown coefficients which can be found from the boundary conditions. In correspondence with Uno et al. (1989), we take the initial and terminal velocities and accelerations to be zero: hQ (!¹ )"hQ (¹ )"h® (!¹ )"h® (¹ )"0

(8)

For mathematical convenience, we have chosen the time at movement onset and movement termination as !¹ and ¹, respectively. For the same reason, and without loss of generality — the minimum-torque-change cost function (3) is independent of angular position and therefore exhibits translation invariance — we impose the following conditions for the arm angular position: h(¹ )"!h(!¹ )"h (9) f Insertion of these conditions into (7) and straightforward algebraic manipulation yield Q(u) 1 h u "h , u "! f Q(u)u2 1 f sinh u 3 6 ¹3

A

B

1 h u " f Q(u) u2!u coth u 5 ¹ 2

u "u "u "0 2 4 6 for the coefficients, with u,b¹"B¹/I and

(10)

Q(u),3/(u2!3u coth u#3)

(11)

With these we obtain the minimum-torque-change trajectory equation

A

AB B B

sinh bt 1 t 3 h(t)"h Q(u) ! u2 f sinh u 6 ¹

A

#

1 t u2!u coth u 2 ¹

(12)

In order to facilitate comparisons between trajectories of various durations, we rewrite (12) in time-normalized form as

A

A

BB

sinh uq 1 1 h(q)"h Q(u) ! u2 q3# u2!u coth u q f sinh u 6 2 q,t/¹3[!1, 1]

(13)

The corresponding time-normalized velocity equation is d h(q)"¹ hQ (q) dq "h Q(u) f u cosh uq 1 1 ! u2q2# u2!u coth u ] sinh u 2 2

A

B

(14)

From (13) and (14), two important properties of oneDOF horizontal-plane minimum-torque-change movements follow directly: ¸inear amplitude scaling and velocity profile symmetry; the former results from h f

Fig. 1. Time- and amplitude-normalized minimum-torque-change trajectories (a), velocity profiles (b), and acceleration profiles (c) for u"0.04, 4, and 4000

multiplying all other terms in (13), and the latter follows from the fact that all terms in (14) are even in q. We can also immediately see that the trajectory remains unchanged under the transformation h P!h , qP!q, f f and so it exhibits motion-reversal invariance; with this in mind, we can assume h '0 in what follows. f The dependence of minimum-torque-change trajectories on the parameter u is less straightforward. Consider Fig. 1, which shows time-normalized minimum-torquechange trajectories (Fig. 1a) and the corresponding velocity (Fig. 1b) and acceleration (Fig. 1c) profiles for three different values of u. The lowest of the three values, u"0.04, corresponds to a forearm inertia I"0.1 kg m2, a viscosity B"0.08 kg m2 s~1, and a movement duration 2¹"0.1 s. The inertia and viscosity estimates are taken from Uno et al. (1989) to ensure compatibility with their results. Figure 1 clearly illustrates that the temporal characteristics of minimum-torque-change movements vary as functions of movement duration, viscosity, and inertia. The fact that all three trajectories have identical velocity peak locations suggests, on the other hand, that some trajectory characteristics may be invariant. Below, we will make some more precise statements based on a mathematical analysis of the minimum-torque-change trajectory equation.

324

particular,

3 Properties of planar horizontal minimum-torquechange movements with one degree of freedom In this section, we derive a number of theorems about one-DOF planar horizontal minimum-torque-change movements which are either independent of inertia, viscosity, and movement duration or which apply to certain parameter configurations of particular theoretical interest. In what follows, we will use the term ‘minimumtorque-change movement’ to mean any one-DOF planar horizontal movement that obeys (13). The first theorem addresses the issue of trajectory equivalence. Theorem 1. ¹wo minimum-torque-change movements with parameter configurations u"B¹/I and u@"B@¹@/I@ have identical time-normalized trajectories if and only if u"u@. Theorem 1 contains two statements. The first is that all combinations of viscosity, inertia, and movement duration values for which the product B¹/I is identical have identical time-normalized trajectories. This follows directly from the fact that B, I, and ¹ occur in (13) exclusively in the form of the product B¹/I. The second statement is that two sets of parameters MB, I, ¹N and MB@, I@, ¹ @ N for which u"B¹/IOB@¹ @/I@"u@ never produce identical trajectories. This statement is not selfevident and requires a proof. Proof. Let us consider the fifth derivative of (13) with respect to q evaluated at q"0: d5 h(q) dq5

K

N

= Q(u) "h u5 "3h + b u2k~6 k f sinh u f q/0 k/3

(15)

where b '0 for k*3 (see Appendix). The derivative of k (15) with respect to u,

A

d d5 h(q) du dq5

K B

N

B

= 2 + b u2(k~3) k k/3

B

(17)

Proof. From (13) we obtain

A

u2 Q(u) 3 sinh uq# (3q!q3)sinh u h(q)"h f 3 sinh u 2

B

#3uq cosh u

(18)

Using the small-argument expansion (A4) for Q(u)/3 sinh u and the power series for the hyperbolic functions (Zwillinger 1996), we have

A

B

15 u2 h(q)+h 1! f u5 14

A

= (uq)2k~1 u2 ] 3 + # (3q!q3) (2k!1)! 2 k/1 = u2k~1 = u2k~1 ]+ !3q + (2k!1)! (2k!2)! k/1 k/1 15 u2 = u2 + 3q2k~1! q3 "h 1! f u5 14 2 k/1 3 u2k~1 # u2!6k#3 q 2 (2k!1)!

B

A

B A BB

A

(19)

The limit of (19) for uP0 yields the expected result:

A

1 1 1 q5! q3# q lim h(q)"15h f 40 12 8 u?0

A

B

B

3 5 15 "h q5! q3# q f 8 4 8

(20)

Theorem 3. As uPR, the minimum-torque-change trajectory approaches the shape of a third-order polynomial. In particular,

= "!6h + (k!3)b u2k~7 k f q/0 k/4

A

A

3 5 15 lim h(q)"h q5! q3# q f 8 4 8 u?0

(16)

is strictly negative for all u3R` if h is positive. It f follows that d5h(q)/dq5D is a monotonic function of u, q/0 implying that f: uPd5h(q)/dq5D is one-to-one; thus q/0 uOu@ always implies d5h(q, u)/dq5D Od5 h(q, u@)/ q/0 dq5D and, by extension, h(q, u)Oh(q, u@). q/0 Of particular theoretical interest are parameter combinations MB, I, ¹N for which uP0 or uPR. For uP0, the minimum-torque-change trajectory becomes a fifth-order polynomial, and for uPR it becomes a third-order polynomial. We state these results in the following two theorems. Theorem 2. As uP0, the minimum-torque-change trajectory approaches the shape of a fifth-order polynomial. In

lim h(q)"h (!1 q3#3 q) for q3[!1, 1] 2 f 2 u?=

(21)

Proof. Using the definition (11) of Q(u) and dividing both numerator and denominator by u2, (13) becomes 3h f h(q)" 1!3u~1coth u#3/u2 ]

A

sinh uq 1 1 coth u ! q3# q! q u2 sinh u 6 2 u

B

(22)

Given that lim coth u"1, we can immediately u?= see that the factor outside the brackets will tend to 3h f and that the last term in (22) will vanish. Therefore, we only have to prove that u~2 sinh uq/sinh u also vanishes in this limit for q3[!1, 1]. We shall consider the two cases q3[!1, 0) and q3(0, 1]; the case q"0 is trivial.

325

The three solutions of (27), q "0, $1, can be found V%95 by inspection. We can see that q "0 is the only V%95 solution for !1(q(1 because q2(k~1)!1(0 and u2k~1/(2k#1)!'0 in that interval. If we evaluate the third derivative at those points, we find

For negative values of q, sinh uq e~uq lim "! lim u2 sinh u u2eu u?= u?= 1 "! lim e~u(q`1)"0 (23) u2 u?= since q*!1. Similarly, for positive values of q, we have sinh uq e~uq lim " lim u?= u2 sinh u u?= u2eu 1 " lim eu(q~1)"0 (24) u?= u2 since q)1. The only nonvanishing terms yield the desired result. Interestingly, equations (17) and (21) represent the trajectories which minimize jerk and acceleration, respectively. Theorems 2 and 3 therefore establish that the family of all minimum-torque-change trajectories includes minimum-jerk and minimum-acceleration movements as special cases. We may have guessed Theorem 2 from the fact that B"0Nu"0 and B"0NNQ "Ih® Q. Similarly, we may have expected Theorem 3 from the fact that IP0NuPR and I"0NNQ "Bh® . Both theorems are, however, much more general; they apply to all combinations of B,I,¹ for which the limits uP0 and uPR are reached. The following three theorems differ from the preceding ones in that they state properties of minimum-torquechange trajectories which hold for all values u3R`. By extension, they also hold for all combinations of viscosities, inertias, and movement durations, because R` is closed under multiplication and division; there is no combination of parameters MB,I,¹N3R`3 for which B¹/INR`. The first of these parameter-independent theorems states the invariance of the velocity-peak location. Theorem 4. Minimum-torque-change trajectories have a single velocity maximum which is always located at the movement time midpoint. Proof. We first determine the set of all extremal points for hQ (q) in the interval q3[!1, 1]. A necessary condition for a velocity extremum is ¹2h® (q)"h Q(u)u2(sinh uq csch u!q)"0 f which is satisfied if

(25)

sinh uq!q sinh u"0.

(26)

Next, we replace the terms sinh uq and sinh u with their power-series expansions: = (uq)2k~1 = u2k~1q 0" + !+ (2k#1)! (2k#1)! k/1 k/1 = q2(k~1)!1 "q + u2k~1 (2k#1)! k/1

(27)

¹3h® Q (0)"h Q(u)u2 csch u(u!sinh u)(0 f

(28)

¹3h® Q ($1)"h Q(u)u3 csch u(cosh u!1)'0 f Hence the velocity reaches a maximum at the midpoint, as was to be shown; at the movement endpoints, it attains minima, as expected from the boundary conditions (8). Another important property of minimum-torquechange trajectories is that the ratio of peak-velocity to average-velocity is restricted to a small range of values for all possible sets of arm-dynamical parameters. The lower and upper boundaries for that ratio are the peakvelocity to average-velocity ratios of minimum-acceleration movements (3/2) and minimum-jerk movements (15/8), respectively. Theorem 5. ¹he peak-velocity to average-velocity ratio, R(u),h0 /h0 "¹h0 /h , obeys 1%!, !7' 1%!, f 3 15 (R(u)( for all u3R` 2 8

(29)

Proof. From Theorem 4 we know that h0 "h0 (q)D , 1%!, q/0 and hence

A

B

Q(u) 3 R(u)" 3u# u2 sinh u!3u cosh u 3 sinh u 2

(30)

where the term outside the brackets is always positive (see Appendix). We now prove the two statements R(u)!3/2'0 and R(u)!15/8(0 separately. First, consider 3 1 Q(u) R(u)! " (6u#3u cosh u!9 sinh u) 2 2 3 sinh u 1 Q(u) " ] 2 3 sinh u

A

A

B

B

1 = 3 6u#3 + ! u2k~1 (2k!2)! (2k!1)! k/1

A

B

Q(u) = (k!2) " u# + u2k~1 sinh u (2k!1)! k/1

(31)

where we have used the power-series expansions for the hyperbolic functions (Zwillinger 1996). The first term in the series is !u, and the second one vanishes. Every remaining term is positive and thus 3 Q(u) = (k!2) R(u)! " + u2k~1'0 2 sinh u (2k!1)! k/3

(32)

326

Next, we consider

and

15 3 Q(u) R(u)! " (8u!u2 sinh u 8 8 3 sinh u

sinh u 1 1 ' 1' cosh~1 u u J3

A

A

= u2k`1 3 Q(u) 8u! + " (2k!1)! 8 3 sinh u k/1

B

B

cosh u'u~1 sinh u'cosh(u/J3)

B

7 = 15 #+ ! u2k~1 (2k!2)! (2k!1)! k/1 1 1 Q(u) = 7 + "! ! (2k!3)! (2k!2)! 8 sinh u k/2 15 # u2k~1 (2k!1)!

A

= u2k~2 = u2k~2 + (2k!1)' + (2k!1)! (2k!1)! k/1 k/1 (33)

(34)

since the coefficients are positive for k*4. Finally, we consider an important property of minimum-torque-change acceleration profiles. Theorem 6. Minimum-torque-change trajectories have exactly two acceleration extrema. ¹he extrema are restricted to lie in the following intervals: !1(q (!1/J3 and 1/J3(q (1 A%95,1 A%95,2

(35)

Theorem 6 states that the first peak of any minimumtorque-change acceleration profile must occur before 21.13% of the total movement time has elapsed, and the second peak cannot occur before 78.87% of the total movement time has elapsed. Proof. A necessary condition for an extremum of h® (q) is ¹3h® Q (q)"h Q(u)u2(u csch u cosh uq!1)"0 f or simply

(36)

cosh uq"u~1 sinh u for u'0

(37)

Solving for q, we obtain

A

1 sinh u q "$ cosh~1 A%95,1,2 u u

B

(38)

The locations of the two acceleration extrema. We can now restate the theorem as follows:

A

B

sinh u 1 1 (! !1(! cosh~1 u u J3

(41)

or, in power-series form,

where we renamed the dummy index in the first series and cancelled the first term in the second one with 8u. Now, 15 1 Q(u) = u2k~1 + R(u)! "! (2k!1)! 8 2 sinh u k/4 ](k!2)(k!3)(0

(40)

Since the two expressions only differ by a factor of !1, it is sufficient to prove the positive case. The hyperbolic cosine is one-to-one for positive arguments, and thus a statement equivalent to (40) is

#7u cosh u!15 sinh u)

A

B

(39)

= u2k~2 '+ 31~k(2k!1) (42) (2k!1)! k/1 Since 2k!1"1 for k"1 and 2k!1'1 for k*2, we have cosh u'u~1 sinh u. Similarly, from 31~k(2k!1)"1 for k"1, 2 and 31~k(2k!1)(1 for k*3 we conclude that u~1 sinh u'cosh 3~1@2 u. 4 Discussion We may divide the six theorems into two groups: (1) Parameter-dependent theorems (Theorems 1 through 3) and (2) parameter-independent theorems (Theorems 4 through 6). The parameter-dependent theorems refer to properties of minimum-torque-change trajectories which hold for particular configurations of movement duration, viscosity, and inertia. We will discuss these first. Theorem 1 establishes that the identity of the product B¹/I is a necessary and sufficient condition for trajectory equivalence. The following predictions can be derived from that statement: (1) A change in the product of viscosity and movement duration will not alter the timenormalized minimum-torque-change trajectory if the inertia is changed by the same factor; (2) a change in viscosity by some factor will not affect the timenormalized minimum-torque-change trajectory if the movement duration is changed by the inverse of that factor; (3) all other changes in movement duration, viscosity, and inertia will alter the time-normalized minimum-torque-change trajectory. Theorems 2 and 3 establish the connection between minimum-torque-change trajectories and both minimum-jerk and minimum-acceleration trajectories. Theorem 2 states that minimum-torque-change trajectories approach the shape of minimum-jerk trajectories as uP0. For u"0.04 (see Fig. 1), the minimumtorque-change trajectory has a ratio of peak to average velocity of R"1.87499 and acceleration peaks at q "$0.57736. Since these values differ from the corA%95 responding minimum-jerk values by only 0.00001, we can take u"0.04 as a good approximation to uP0. It follows that fast minimum-torque-change movements are essentially indistinguishable from the corresponding

327

minimum-jerk movements, provided Uno et al.’s (1989) estimates for inertia and viscosity are accurate. Theorem 3 states that minimum-torque-change trajectories approach the shape of minimum-acceleration trajectories as uPR. A sufficiently accurate approximation for that limiting value is obtained by letting u"4000 (see Fig. 1), for which the values R"1.50037 and q "$0.992926 closely match the corresponding A%95 minimum-acceleration theoretical values of R"3/2 and q "$1. Experimentally, we can attain u values of A%95 this magnitude by having subjects perform slow movements against a large viscous force. Although Theorems 1 through 3 are experimentally testable, certain complications arise from their parameter dependency. Currently used methods for estimating inertia (Contini et al. 1963; Hinrichs 1985; Jensen 1978) and viscosity (Lacquaniti et al. 1982) are rather crude, and little is known about their accuracy. Discrepancies between model predictions and experimental data may therefore be the result of an inaccurate model or inaccurate parameter estimates. Conclusive inferences about the validity of the minimum-torque-change principle cannot therefore be based on Theorems 1 through 3. These testing problems do not apply to Theorems 4 through 6. They make predictions which hold for all values of movement duration, viscosity, and inertia, and several experimental studies actually report results which can be directly compared with these predictions. It is necessary to recall, however, that all theorems apply exclusively to horizontal-plane movements with one mechanical DOF. We have found four studies (Baba and Marteniuk 1983; Moore and Marteniuk 1986; Nagasaki 1989; Wiegner and Wierzbicka 1992) which satisfy these requirements and which report data that are directly related to our results. Theorem 4 states that minimum-torque-change trajectories have a single velocity peak which is always located at the movement time midpoint. Three of the four single-DOF studies we found report velocity peak locations; none of these is fully compatible with Theorem 4. Moore and Marteniuk (1986) and Nagasaki (1989) both investigate forearm-flexion movements of various durations and find that slow movements reach peak velocity after less than 50% of total movement time, whereas fast movements reach peak velocity around the movement midpoint (Moore and Marteniuk 1986) or even later (Nagasaki 1989). A third study by Wiegner and Wierzbicka (1992) shows that extremely fast elbow-flexion movements reach peak velocity after as much as 58% of the total duration (standard error"3%). Taken together, these studies indicate that the ratio of velocitypeak location to movement duration is a function of the latter and therefore not an invariant. This is a clear violation of Theorem 4. Two of the above studies also report the ratios of peak to average velocities (R), and again these findings are at odds with the minimum-torque-change model (Theorem 5). Nagasaki (1989) finds that slow movements have R values between 1.77 and 1.89 for various amplitude conditions, while fast movements have R values

between 2.01 and 2.09. The latter are similar to Wiegner and Wierzbicka’s (1992) result of an average R value of 2.13 for movements which are slightly faster than those in Nagasaki’s fastest condition. These data indicate that Theorem 5, which predicts a maximum R value of 15/8"1.875, may not hold for fast forearm flexions. Two of the four one-DOF studies we considered also report the relative acceleration-peak locations. Here the results are mixed. Baba and Marteniuk (1983) find acceleration-peak locations at 33.3%—35.5% of total movement time for loaded and unloaded fast forearm flexions. These results are at variance with Theorem 6, which predicts that the first acceleration extremum of a minimum-torque-change movement must occur after less than 21.13% of total movement time. Moore and Marteniuk (1986), on the other hand, find acceleration peaks at 8.3% and 12.7% of total movement time for fast movements with and without practice, respectively, and at 7.6% and 6.6% for the corresponding slow movements; these numbers are within the limits specified by Theorem 6.

5 Conclusions In this paper, we have derived various properties of one-DOF horizontal-plane movements. At the heart of this analysis are the parameter-independent properties stated in Theorems 4 through 6. The particular strength of these theorems is that they result in predictions which are independent of any segment-parameter estimates; they must hold for any combination of movement duration, viscosity, and inertia values. Several studies were re-analyzed in the light of these new results and were found to contradict the model’s predictions concerning the velocity-peak location and the ratio of peak velocity to average velocity. The observation that slow movements are skewed to the right — i.e., that they show a velocity peak after less than 50% of the movement duration — may be explained as the result of the superposition of a major ballistic movement component and one or several corrective components (Milner 1992; Milner and Ijaz 1990; Berthier 1996).1 If each component has a single velocity peak located at the movement-time midpoint, the minimum-torque-change model may still be valid for these submovements, albeit not for the composite movement. Note, however, that the reverse asymmetry observed in fast movements cannot be explained in this fashion. An even more serious challenge for the minimumtorque-change model is the finding that the ratio of peak to average velocity of fast forearm flexions significantly exceeds the predicted upper limit of 15/8. Confirmation of this finding may seriously limit the model’s validity.

1 Fast movements might not have corrective components because visual reaction time may exceed movement duration. See Schmidt (1979) for a review.

328 Acknowledgements. This research was supported in part by the German Academic Exchange Service (DAAD) through an HSPII/AUFE PhD scholarship to S.E.E. We thank A. G. Barto, N. E. Berthier, M. H. Fischer, L. R. Hunter, L. L. Loukopoulos, and E. Navayazdani for helpful comments on an earlier draft of this paper.

Appendix From the definition (11), we have 3 sinh u "u2 sinh u!3u cosh u#3 sinh u Q(u)

(A1)

We now use the power-series expansions for the hyperbolic functions (Zwillinger 1996) to obtain 3 sinh u = u2k~1 = u2k~2 = u2k~1 "u2 + !3u + #3 + Q(u) (2k!1)! (2k!2)! (2k!1)! k/1 k/1 k/1 = u2k`1 = 1 1 "+ #3 + ! u2k~1 (2k!1)! (2k!1)! (2k!2)! k/1 k/1 = u2k~1 = 1 1 "+ #3 + ! u2k~1 (A2) (2k!3)! (2k!1)! (2k!2)! k/2 k/2 where we renamed the dummy index in the first sum and used the fact that 0!"1!"1 in the second. Now, the coefficient for the k"2 term in (A2) vanishes, and hence

A A

B B

A

B

= 1 3 3 3 sinh u "+ # ! u2k~1 (2k!3)! (2k!1)! (2k!2)! Q(u) k/3 = 1 "+ ((2k!2)(2k!1)#3!3(2k!1))u2k~1 (2k!1)! k/3 = (k!2)(k!1) = "4 + u2k~1, + b u2k~1 (A3) k (2k!1)! k/3 k/3 Each coefficient b thus defined is clearly positive for k*3. For small k values of u,

A

B

3 sinh u u5 u7 u5 u2 + # " 1# Q(u) 15 210 15 14

(A4)

and

A

B

15 u2 Q(u) + 1! 14 3 sinh u u5

(A5)

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