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

Transverse linearization-based approaches have become among the most prominent methods for orbitally stabilizing feedback design in regards to (periodic) motions of underactuated mechanical systems. Yet, in an $n$-dimensional state-space, this requires knowledge of a set of $(n-1)$ independent transverse coordinates, which can be nontrivial to find and whose definitions might vary for different motions (trajectories). In this paper, we consider instead a generic set of $excessive$ transverse coordinates which are 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 transverse linearization, as well as state a sufficient condition for the existence of a feedback controller rendering the desired trajectory (locally) asymptotically orbitally stable. The presented approach is applied to stabilizing oscillations of the underactuated cart-pendulum system about its unstable upright position, in which a novel motion planning approach based on virtual constraints is utilized for trajectory generation.