THE DRIVEN ROLLING TORUS

We develop the equations of motion for a torus with a general contact profile that rolls without slipping and that is actuated using drive and steer torques. The form of the dynamics plus nonholonomic constaints allows an easy reduction to a dynamic control system of dimension 4 including the three angular rates and the roll (or lean) angle. Using a standard torus with geometric and inertial parameters similar to those of a motorcycle tire and wheel, we find that the rolling torus exhibits, with increasing forward speed, an unexpected transition from stability to instability and back to stability.