  This paper applies geometric PID control for asymptotic tracking of a desired
trajectory by a hoop robot in the presence of disturbances and uncertainties.
The hoop robot, consisting of a circular body rolling without slip along a
one-dimensional surface, is a planar analog of a spherical robot. A variety of
coupled mechanical system may be used to actuate the hoop robot. This paper
specifically considers two different actuators, one a simple pendulum and the
other an internal cart. The geometric PID controller requires the plant to be a
mechanical system, and the hoop robot does not satisfy this condition.
Therefore a geometric inner loop is presented that gives the hoop robot the
required structure. This procedure is here referred to as feedback
regularization. Feedback regularization--in contrast to feedback
linearization--is coordinate independent, and hence reflects the fundamental
system structure. Note also that the resulting mechanical system is nonlinear
and underactuated. Subsequently, the geometric PID outer loop guarantees
almost-semiglobal tracking with locally exponential convergence, and the
integral action of the PID guarantees robustness to constant disturbances and
parameter uncertainties, including constant inclination of the rolling surface.
The complete tracking controller is the composition of the two
coordinate-independent loops, and therefore is also coordinate independent.
