Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems

In this paper we develop a constructive approach to the determination of stabilizing control laws for a class of Lagrangian mechanical systems with symmetry - systems whose underlying dynamics are governed by the Euler-Poincaré equations. This work extends our previous work on the stabilization of mechanical control systems using the method of controlled Lagrangians. The guiding principle behind our methodology is to develop a class of stabilizing feedback control laws which yield closed-loop dynamics that remain in Lagrangian form. Using the methodology for Euler-Poincaré systems, we analyse stabilization of a satellite and an underwater vehicle controlled with momentum wheels.