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In this paper, we present a sliding mode control algorithm to robustly stabilize a class of underactuated mechanical systems that are not linearly controllable and violate Brockett's necessary condition for smooth asymptotic stabilization of the equilibrium, with parametric uncertainties. In defining the class of systems, a few simplifying assumptions are made on the structure of the dynamics; in particular, the damping forces are assumed to be linear in velocities. We first propose a switching surface design for this class of systems, and subsequently, a switched algorithm to reach this surface in finite time using conventional and higher order sliding mode controllers. The stability of the closed-loop system is investigated with an undefined relative degree of the sliding functions. The controller gains are designed such that the controller stabilizes the actual system with parametric uncertainty. The proposed control algorithm is applied to two benchmark problems: a mobile robot and an underactuated underwater vehicle. Simulation results are presented to validate the proposed scheme.
Sankaranarayanan et al. (Fri,) studied this question.