  A spherical robot consists of an externally spherical rigid body rolling on a
two-dimensional surface, actuated by an auxiliary mechanism. For a class of
actuation mechanisms, we derive a controller for the geometric center of the
sphere to asymptotically track any sufficiently smooth reference trajectory,
with robustness to bounded, constant uncertainties in the inertial properties
of the sphere and actuation mechanism, and to constant disturbance forces
including, for example, from constant inclination of the rolling surface. The
sphere and actuator are modeled as distinct systems, coupled by reaction
forces. It is assumed that the actuator can provide three independent control
torques, and that the actuator center of mass remains at a constant distance
from the geometric center of the sphere. We show that a necessary and
sufficient condition for such a controller to exist is that for any constant
disturbance torque acting on the sphere there is a constant input such that the
sphere and the actuator mechanism has a stable relative equilibrium. A
geometric PID controller guarantees robust, semi-global, locally exponential
stability for the position tracking error of the geometric center of the
sphere, while ensuring that actuator velocities are bounded.
