Forced nonlinear oscillation of underactuated systems: the balance board example

Abstract

In this contribution we consider a mechanical model of the so-called balance board or "indoboard" system, which is an unstable underactuated mechanical system. Instead of stabilizing one equilibrium point we investigate the asymptotic stabilization of a closed orbit, which results in a nonlinear oscillation. To this end the system is considered in coordinates of the real Jordan normal form, which facilitates the construction of a Control-Lyapunov function. On this basis, a nonlinear feedback law which renders the periodic orbit asymptotically stable is easily derived via Sontags formula.