Prioritized Inverse Kinematics: Nonsmoothness, Trajectory Existence, Task Convergence, Stability

In this paper, we study various theoretical properties of a class of prioritized inverse kinematics (PIK) solutions that can be considered as a class of (regulation or output tracking) control laws of a dynamical system with prioritized multiple outputs. We first develop tools to investigate nonsmoothness of PIK solutions and find a sufficient condition for nonsmoothness. It implies that existence and uniqueness of a joint trajectory satisfying the PIK solution cannot be guaranteed by the classical theorems. So, we construct an alternative existence and uniqueness theorem that uses structural information of the PIK solution. Then, we narrow the class of PIK solutions down to the case that all tasks are designed to follow some desired task trajectories and discover a few properties related to convergence. The study goes further to analyze stability of equilibrium points of the differential equation whose right hand side is the PIK solution when all tasks are designed to reach some desired task positions. Finally, we furnish an example with a two-link manipulator that shows how our findings can be used to analyze the behavior of the joint trajectory generated from the PIK solution.