  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.
