Excessive Transverse Coordinates for Orbital Stabilization of (Underactuated) Mechanical Systems

Abstract : The transverse linearization is a useful tool for the design of feedback controllers that orbitally stabilizes (periodic) motions of mechanical systems. Yet, in an n-dimensional state-space, this requires knowledge of a set of (n − 1) independent transverse coordinates, which at times can be difficult to find and whose definitions might vary for different motions (trajectories). Motivated by this, we present in this paper a generic choice of excessive transverse coordinates defined in terms of a particular parameterization of the motion and a projection operator recovering the "position" along the orbit. We present a constructive procedure for obtaining the corresponding excessive transverse linearization and state a sufficient condition for the existence of a feedback controller rendering the desired trajectory (locally) asymptotically orbitally stable. The approach is demonstrated through numerical simulation by stabilizing oscillations around the unstable upright position of the underactuated cart-pendulum system, in which a novel motion planning approach based on virtual constraints is utilized for trajectory generation.