Determining the principles used to plan and execute movements is a fundamental question in neuroscience research. When humans reach to a target with their hand, they exhibit stereotypical movements that closely follow an optimally smooth trajectory. Even when faced with various perceptual or mechanical perturbations, subjects readily adapt their motor output to preserve this stereotypical trajectory. When humans manipulate non-rigid objects, however, they must control the movements of the object as well as the hand. Such tasks impose a fundamentally different control problem than that of moving one's arm alone. Here, we developed a mathematical model for transporting a mass-on-a-spring to a target in an optimally smooth way. We demonstrate that the well-known "minimum-jerk" model for smooth reaching movements cannot accomplish this task. Our model extends the concept of smoothness to allow for the control of non-rigid objects. Although our model makes some predictions that are similar to minimum jerk, it predicts distinctly different optimal trajectories in several specific cases. In particular, when the relative speed of the movement becomes fast enough or when the object stiffness becomes small enough, the model predicts that subjects will transition from a uni-phasic hand motion to a bi-phasic hand motion. We directly tested these predictions in human subjects. Our subjects adopted trajectories that were well-predicted by our model, including all of the predicted transitions between uni- and bi-phasic hand motions. These findings suggest that smoothness of motion is a general principle of movement planning that extends beyond the control of hand trajectories.
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