Origins and violations of the 2/3 power law in rhythmic three-dimensional arm movements

The 2/3 power law, the nonlinear relationship between tangential velocity and radius of curvature of the end-effector trajectory, is thought to be a fundamental constraint of the central nervous system in the formation of rhythmic endpoint trajectories. However, studies on the 2/3 power law have been confined largely to planar drawing patterns of relatively small size. With the hypothesis that this strategy overlooks nonlinear effects that are constitutive in movement generation, the present experiments tested the validity of the power law in elliptical patterns that were not confined to a planar surface and which were performed by the unconstrained 7-degrees of freedom (DOF) arm, with significant variations in pattern size and workspace orientation. Data were recorded from five human subjects where the seven joint angles and the endpoint trajectories were analyzed. Additionally, an anthropomorphic 7-DOF robot arm served as a "control subject" whose endpoint trajectories were generated on the basis of the human joint angle data, modeled as simple harmonic oscillations. Analyses of the endpoint trajectories demonstrate that the power law is systematically violated with increasing pattern size, in both exponent and the goodness of fit. The origins of these violations can be explained analytically based on smooth, rhythmic trajectory formation and the kinematic structure of the human arm. We conclude that, in unconstrained rhythmic movements, the power law seems to be a by-product of a movement system that favors smooth trajectories, and that it is unlikely to serve as a primary movement-generating principle. Our data rather suggest that subjects employed smooth oscillatory pattern generators in joint space to realize the required movement patterns.