Stable manifolds of saddle equilibria for pendulum dynamics on S2 and SO(3)

Global nonlinear dynamics of various classes of closed loop attitude control systems have been studied in recent years [1]. Closely related results on attitude control of a spherical pendulum (with attitude an element of the twosphere S2) and of a 3D pendulum (with attitude an element of the special orthogonal group SO(3)) are given in [2], [3]. These publications address the global closed dynamics of smooth vector fields on nonlinear manifolds. Assuming that the controlled system has an asymptotically stable equilibrium, as desired in attitude stabilization problems, additional hyperbolic equilibria necessarily appear [4]. As a result, the desired equilibrium is not globally asymptotically stable, since the domain of attraction of the desired asymptotically stable equilibrium excludes the union of the stable manifolds of the hyperbolic equilibria. It is referred to as almost globally asymptotically stable, as the stable manifolds of the hyperbolic equilibria have lower dimension than the attitude configuration manifold. However, the characteristics of the stable manifolds to the hyperbolic equilibria and the corresponding effects on the solutions have not been directly studied in the prior literatures. These geometric factors motivate the current paper, in which new computational results to visualize the stable manifolds of the hyperbolic equilibria are developed. To make the development concrete, the presentation is built around two specific closed loop vector fields: one for the