On Almost-Global Tracking for a Certain Class of Simple Mechanical Systems

In this paper, we propose a control law for the almost-global asymptotic tracking (AGAT) of a smooth reference trajectory for a fully actuated simple mechanical system (SMS) evolving on a Riemannian manifold that 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 the 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 ℝ
³
.