Asymptotic Stabilization of the Inverted Equilibrium Manifold of the 3-D Pendulum Using Non-Smooth Feedback

The 3-D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. In [6], almost global stabilization of the inverted equilibrium manifold was studied using a smooth globally defined feedback. Here, we study the problem of almost global stabilization of the inverted equilibrium manifold using non-smooth feedback of angular velocity and a reduced attitude vector of the 3-D pendulum. The importance of the non-smooth feedback is that the almost global domain of attraction is a geometrically simple set that excludes the hanging attitude manifold. Unlike the closed-loop for a 3-D pendulum with a smooth controller, the closed-loop designed in this paper does not exhibit a performance constraint. These new results are based on Lyapunov analysis of the non-smooth closed-loop 3-D pendulum.