According to the neuromuscular model of virtual trajectory control, the postures and movements of limbs are performed by shifting the equilibrium positions determined by agonist and antagonist muscle activities. In this study, we develop virtual trajectory control for the reaching movements of a multi-joint arm, introducing a proportional-derivative feedback control scheme. In virtual trajectory control, it is crucial to design a suitable virtual trajectory such that the desired trajectory can be realized. To this end, we propose an algorithm for updating virtual trajectories in repetitive control, which can be regarded as a Newton-like method in a function space. In our repetitive control, the virtual trajectory is corrected without explicit calculation of the arm dynamics, and the actual trajectory converges to the desired trajectory. Using computer simulations, we assessed the proposed repetitive control for the trajectory tracking of a two-link arm. Our results confirmed that when the feedback gains were reasonably high and the sampling time was sufficiently small, the virtual trajectory was adequately updated, and the desired trajectory was almost achieved within approximately 10 iterative trials. We also propose a method for modifying the virtual trajectory to ensure that the formation of the actual trajectory is identical even when the feedback gains are changed. This modification method makes it possible to execute flexible control, in which the feedback gains are effectively altered according to motion tasks.

It is well known that planar reaching movements of the human shoulder and elbow joints have invariant features: roughly straight hand paths and bell-shaped velocity profiles. The optimal control models with the criteria of smoothness or precision, which determine a unique movement pattern, predict such features of hand trajectories. In this letter on expanding the research on simple arm reaching movements, we examine whether the smoothness criteria can be applied to whole-body reaching movements with many degrees of freedom. Determining a suitable joint trajectory in the whole-body reaching movement corresponds to the optimization problem with constraints, since body balance must be maintained during a motion task. First, we measured human joint trajectories and ground reaction forces during whole-body reaching movements, and confirmed that subjects formed similar movements with common characteristics in the trajectories of the hand position and body center of mass. Second, we calculated the optimal trajectories according to the criteria of torque and muscle-tension smoothness. While the minimum torque change trajectories were not consistent with the experimental data, the minimum muscle-tension change model was able to predict the stereotyped features of the measured trajectories. To explore the dominant effects of the extension from the torque change to the muscle-tension change, we introduced a weighted torque change cost function. Considering the maximum voluntary contraction (MVC) force of the muscle as the weighting factor of each joint torque, we formulated the weighted torque change cost as a simplified version of the minimum muscle-tension change cost. The trajectories owing to the minimum weighted torque change criterion also showed qualitative agreement with the common features of the measured data. Proper estimation of the MVC forces in the body joints is essential to reproduce human whole-body movements according to the minimum muscle-tension change criterion.