Analysis and Control of Unstable Rolling Wheel Dynamics

Many vehicle systems contain rolling elements which exhibit unstable rolling motion, called shimmying, which may lead to disastrous results. The classical shimmying wheel is a simple model which captures the essential dynamics of such systems. In particular, the uncontrolled stability of the rolling wheel is characterized by a subcritical Hopf bifurcation when certain parameter values are varied. As such, the equilibrium state for the system may be unstable, and even if it is stable, sufficiently non-local initial conditions will lead to unstable dynamics. To control this unstable behavior, a particular geometric structure of this model can be exploited which provides a simple means to design a globally stabilizing controller by the means of feedback linearization.