Orbital Stabilization of Periodic Motions Induced by Singular Virtual Holonomic Constraints

This paper addresses the problem of orbital stabilization of periodic trajectories in underactuated mechanical systems. We focus on cases where the desired trajectory arises from a nonregular virtual holonomic constraint, leading to reduced dynamics with isolated singularities. Existing stabilization methods are generally inapplicable or ineffective in such cases.
We propose an alternative approach based on the approximate linearization of the transverse dynamics, which yields a linear time-varying system that remains regular at singular points and is typically controllable. A stabilizing LQR-based feedback is then designed for this linear system, and we show that it ensures local orbital asymptotic stability of the original nonlinear system’s trajectory.
The effectiveness of the proposed approach is illustrated through the stabilization of a periodic trajectory in the Pendubot system, where the second link oscillates around the horizontal position.