Almost-globally stable tracking for on compact Riemannian manifolds
It solves the tracking problem for mechanical systems on general compact Riemannian manifolds, which was previously only local or restricted to Lie groups.
The paper proposes a control law achieving almost-global asymptotic tracking for fully actuated simple mechanical systems on compact Riemannian manifolds embeddable in Euclidean space, extending prior results beyond Lie groups. The method is demonstrated on a spherical pendulum and a particle on a Lissajous curve.
In this article, we propose a control law for almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving on a Riemannian manifold which can be embedded in a Euclidean space. The existing results on tracking for an SMS are either local, or almost-global, only in the case the manifold is a Lie group. In the latter case, the notion of a configuration error is naturally defined by the group operation and facilitates a global analysis. However, such a notion is not intrinsic to a Riemannian manifold. In this paper, we define a configuration error followed by error dynamics on a Riemannian manifold, and then prove AGAT. The results are demonstrated for a spherical pendulum which is an SMS on $S^2$ and for a particle moving on a Lissajous curve in $\mathbb{R}^3$.