This paper describes a systematic procedure to exponentially stabilize relative equilibria of mechanical systems. We review the notion of relative equilibria and their stability in a Riemannian geometry context. Potential shaping and damping control are employed to obtain full exponential stabilization of the desired trajectory. Two necessary conditions are that the effective potential be positive definite over a specified subspace and that the system be linearly controllable. Relevant applications to underwater and aerospace vehicle control are described.
Bullo, F. (2000). Stabilization of relative equilibria for underactuated systems on Riemannian manifolds. Automatica, 36(12), 1819–1834. https://doi.org/10.1016/S0005-1098(00)00115-1