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This paper describes a novel logic-based controller for the global asymptotic stabilization of an underactuated surface vessel, which has a larger degree-of-freedom than the number of control inputs. This system cannot be asymptotically stabilized using any time-invariant smooth feedback controller since the system violates the well-known Brockett's condition (1983). Thus, the proposed controller can be viewed as a time-varying controller devised to circumvent this limitation. It consists of a supervising logic unit and subordinate time-invariant controllers. The asymptotic stability is rigorously proven using Lyapunov analysis, and the practicality of the proposed algorithm is also demonstrated through the simulations of head-in parking and parallel parking.