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.

There is substantial evidence that humans possess an accurate and adaptable internal model of the dynamics of their arm that is utilized by the nervous system for controlling arm movements. However, it is not known if such model-based strategies are used for controlling dynamical systems outside the body. The need to predict events in the external world is not restricted to the execution of reaching movements or to the handling of rigid tools. Model-based control may also be critical for performing functional tasks with non-rigid objects such as stabilizing a cup of coffee. The present study investigated the strategies used by humans to control simple mass-spring objects. Subjects made straight line reaching movements to a target while interacting with a robotic manipulandum that simulated the dynamics of a one-dimensional mass on a spring. After learning, neither hand nor object kinematics returned to those of free reaching, suggesting that this task was not learned as a perturbation of free reaching. Although there are control strategies (such as slowing the movement of the hand) that would require little or no knowledge of object dynamics, subjects did not adopt these strategies. Instead, they tailored their motor commands to the particular object being manipulated. When object parameters were unexpectedly altered in a way that required no changes in kinematics to successfully complete the task, subjects nonetheless exhibited substantial kinematic deviations. These deviations were consistent with those predicted by a model of the arm-plus-object system driven by a low-impedance controller that incorporated an explicit inverse model of arm-plus-object dynamics. The observed behavior could not be reproduced by a controller that relied on modulating hand impedance alone with no inverse model. These results were therefore consistent with the hypothesis that subjects learn to control the kinematics of manipulated objects by forming an internal model that specified the forces to be exerted by the hand on the object to induce the desired motion of that object.

To record three-dimensional coordinates of the joints from normal human subjects during locomotion, we used a digital motion analysis system (ELITE). Recordings were obtained under several different conditions, which included normal walking and stepping over obstacles. Principal component analysis was used to analyze coordinate data after conversion of the data to segmental angles. This technique gave a stable summary of the redundancy in gait kinematic data in the form of reduced variables (principal components). By modeling the shapes of the phase plots of reduced variables (distortion analysis) and using a limited number of model parameters, good resolution was obtained between subtly different conditions. Hence, it was possible to accurately resolve small distributed changes in gait patterns within subjects. These methods seem particularly suited to longitudinal studies in which relevant movement features are not known a priori. Assumptions and neurophysiological applications are discussed.

Joint stiffness measurements during small transient perturbations have suggested that stiffness during movement is different from that observed during posture. These observations are problematic for theories like the classical equilibrium point hypothesis, which suggest that desired trajectories during movement are enforced by joint stiffness. We measured arm impedances during large, slow perturbations to obtain detailed information about the spatial and temporal modulation of stiffness and viscosity during movement. While our measurements of stiffness magnitudes during movement generally agreed with the results of measurements using fast perturbations, they revealed that joint stiffness undergoes stereotyped changes in magnitude and aspect ratio which depend on the direction of movement and show a strong dependence on joint angles. Movement simulations using measured parameters show that the measured modulation of impedance acts as an energy conserving force field to constrain movement. This mechanism allows for a computationally simplified account of the execution of multijoint movement. While our measurements do not rule out a role for afferent feedback in force generation, the observed stereotyped restoring forces can allow a dramatic relaxation of the accuracy requirements for forces generated by other control mechanisms, such as inverse dynamical models.