Sang-ik An

Sang-ik An, Dongheui Lee

A prioritized inverse kinematics (PIK) solution can be considered as a (regulation or output tracking) control law of a dynamical system with prioritized multiple outputs. We propose a method that guarantees that a joint trajectory generated from a class of PIK solutions exists uniquely in a nonsingular configuration space. We start by assuming that desired task trajectories stay in nonsingular task spaces and find conditions for task trajectories to stay in a neighborhood of desired task trajectories in which we can guarantee existence and uniqueness of a joint trajectory in a nonsingular configuration space. Based on this result, we find a sufficient condition for task convergence and analyze various stability notions such as stability, uniform stability, uniform asymptotic stability, and exponential stability in both continuous and discrete times. We discuss why the number of tasks is limited in discrete time and show how preconditioning can be used in order to overcome this limitation.

Sang-ik An, Dongheui Lee

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 (output regulation or 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 a PIK solution cannot be guaranteed by the classical theorems. So, we construct an alternative existence and uniqueness theorem that uses structural information of PIK solutions. 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 task convergence. The study goes further to analyze stability of equilibrium points of the differential equation whose right hand side is a 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 a joint trajectory generated from a PIK solution.