Attitude stabilization of the inverted 3D pendulum on TSO(3) with control saturation

This paper treats the asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold of the 3D pendulum, taking into account saturation of the control moment. Control saturation is accommodated by a novel feedback structure that is based on the special geometric features of the 3D pendulum, namely that the attitude representation lies in the compact manifold SO(3). This attitude stabilization problem is solved by use of Lyapunov methods applied to closed loop dynamics that evolve on the tangent bundle TSO(3). The construction of a Lyapunov function for the closed-loop, that explicitly involves saturation, exemplifies a method to handle saturation for attitude stabilization on SO(3) when potential forces exist. The controller provides freedom to influence the local dynamics of the closed loop near the specified equilibrium and the global dynamics away from the equilibrium, as well as some freedom to shape the manifold of solutions that do not converge to the specified equilibrium.