Stabilization of a Specified Equilibrium in the Inverted Equilibrium Manifold of the 3D Pendulum

This paper treats the asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold of the 3D pendulum. This attitude stabilization problem is solved by use of Lyapunov methods applied to closed loop dynamics that evolve on the tangent bundle TSO (3). A smooth controller is proposed that achieves almost global asymptotic stabilization of the specified equilibrium; the controller provides freedom to influence the local dynamics of the closed loop near the specified equilibrium as well as some freedom to shape the manifold of solutions that do not converge to the specified equilibrium.