Geometric PID Controller for Stabilization of Nonholonomic Mechanical Systems on Lie Groups

The PID controller is an elegant and versatile controller for set point tracking in double integrator systems of which mechanical systems evolving on Euclidean space constitute a large class. But since mechanical systems are typically constrained interconnections of rigid bodies whose configuration space is $SE(3)$, which is not even topologically Euclidean, a geometric PID controller has been developed for mechanical systems evolving on Lie groups. In this work, we extend the framework to such systems which have nonholonomic constraints. It encompasses many practically applicable mechanical systems encountered in robotics as robots are constrained interconnections of rigid bodies where the constraints could either be holonomic or nonholonomic.