Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis

Abstract

The movements of the human arm have been extensively studied for a variety of goal-directed experimental tasks. Analyses of the trajectory and velocity of the arm have led to many hypotheses for the planning strategies that the CNS might use. One family of control hypotheses, including minimum jerk, snap and their generalizations to higher orders, comprises those that favor smooth movements through the optimization of an integral cost function. The predictions of each order of this family are examined for two standard experimental tasks: point-to-point movements and the periodic tracing of figural forms, and compared both with experiment and the two-thirds power law. The aim of the analyses is to generalize previous numerical observations as well as to examine movement segmentation. It is first shown that contrary to recent statements in the literature, the only members of this family of control theories that match reaching movement experiments well are minimum jerk and snap. Then, for the case of periodic drawing, both the ellipse and cloverleaf are examined and the experimentally observed power law is derived from a first-principles approach. The results for the ellipse are particularly general, representing a unification of the two-thirds power law and smoothness hypotheses for ellipses of all reasonable eccentricities. For complex shapes it is shown that velocity profiles derived from the cost-function approach exhibit the same experimental features that were interpreted as segmented control by the CNS. Because the cost function contains no explicit segmented control, this result casts doubt on such an interpretation of the experimental data.

Fig. 1.

The MSD profiles for n = 2, 3, 4, and 10 with the only experimental input being the time scale T (given by the experimental data). The profiles become narrower and taller as n increases. The experimental curve is best fitted by minimum jerk n = 3 and snap n = 4.

Fig. 2.

The same profiles as in Figure 1 but rescaled as by Harris (1998). Each profile is rescaled such that they all pass through the points marked by arrows. A cut-off for the start and end of the movements must also be supplied.

Fig. 3.

Examples of the generalized cloverleaf for various values of the perturbation parameter ε. See Results (Eq. 11) for the mathematical definition.

Fig. 4.

The log(curvature) versus log(velocity) curve as derived numerically from the minimum-jerk cost function for a cloverleaf with ε = 1 (in this case the units are arbitrary as the fit is to theory, not experiment). The value of the β exponent measured from this curve is β = 0.36, reproducing the results found in the experiment.

Fig. 5.

A plot of the figure-of-eight template. The numerical derivation of the minimum-jerk trajectory was obtained using 10 via-points. The labels M1, M2, and M3 mark the curvature maxima, whereas m1 is the curvature minimum. It is these extrema and their relation by symmetries that determine the number of apparent segments (Fig. 6).

Fig. 6.

The predicted minimum-jerk velocity versus effective radius of curvature plot, for the tracing of the figure-of-eight. The positions of the curvature extrema (M1, M2, M3, and m1) are marked and can be compared with Figure5. The dotted tangential lines give an indication of the gain factors for the two segments (see Results), whereas thedashed lines represent the average experimental values. The theory reproduces the features of segmentation seen in the experimental curves, despite no explicit segmental planning in the structure of the theory.