  These lectures demonstrate the development of a PID control framework for
mechanical systems. Based on the observation that mechanical systems are
essentially double integrator systems, we generalize the linear PID controller
to mechanical systems that have a non-Euclidean configuration space.
Specifically we start by presenting the development of the geometric PID
controller for fully actuated mechanical systems and then extend it to a class
of under actuated interconnected mechanical systems of practical significance
by introducing the notion of feedback regularization. We show that feedback
regularization is the mechanical system equivalent to partial feedback
linearization. We apply these results for trajectory tracking for several
systems of interest in the field of robotics. First, we demonstrate the robust
almost-global stability properties of the geometric PID controller developed
for fully actuated mechanical systems using simulations and experiments on a
multi-rotor-aerial-vehicle. The extension to the class of under actuated
interconnected systems allow one to ensure the semi-almost-global locally
exponential tracking of the geometric center of a spherical robot on an
inclined plane of unknown angle of inclination. The results are demonstrated
using simulations for a hoop rolling on an inclined plane and then for a sphere
rolling on an inclined plane. The final extension that we present here is that
of geometric PID control for holonomically or non-holonomically constrained
mechanical systems on Lie groups. The results are demonstrated by ensuring the
robust almost global locally exponential tracking of a nontrivial spherical
pendulum.
