Geometrical Invariance and Smoothness Maximization for Task-Space Movement Generation

In human motor control studies, end-effector (e.g., hand) trajectories have been successfully modeled using optimization principles. Yet, it remains unclear how such trajectories are updated when the end-effector or task goals are perturbed. Here, we present an approach to human and robotic task-level trajectory planning and modification using geometrical invariance and optimization, allowing to adapt learned movements to a priori unknown boundary conditions. The optimization criterion represents a tradeoff between smoothness (minimum jerk) and accuracy (jerk-accuracy model). We show that planning maximally smooth movements allows recovery from perturbations by superimposing specific affine orbits on maximally smooth preplanned trajectories. The generated trajectories are compared with those resulting from other recent approaches used in robotics. Finally, we discuss conditions for affine invariance of maximally smooth task-space trajectories. Possible applications of this study to both human motor control and robotics research studies are discussed.