Prioritized Inverse Kinematics: Desired Task Trajectories in Nonsingular Task Spaces

Abstract: 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.