  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.
