Modeling and Stabilization for Mechanical Systems

Modeling and Stabilization for Mechanical Systems by Sameer Madhav Jalnapurkar Doctor of Philosophy in Mathematics University of California, Berkeley Professor Jerrold E. Marsden, Chair The main themes of this thesis are modeling and stabilization for mechanical systems. We begin by reviewing how models for mechanical systems can be derived from rst principles, such as Newton's second Law, or balance laws for linear and angular momentum. The Lagrange-d'Alembert equations for modeling constrained systems are shown to follow from such rst principles. Next, we describe how the variational formulation for mechanics can be used to derive equations of motion for the reduced dynamics of a mechanical system with symmetry. The variational principle on the original phase space induces a variational principle on the symplectically reduced space. The next topic is derivation of feedback laws for asymptotic stabilization of relative equilibria of mechanical systems with symmetry. The methodology used is that of potential shaping, but the system is allowed to be underactuated. First, we derive feedback laws to asymptotically stabilize, on a given momentum surface, relative equilibria for which the internal con guration of the system is unstable, but the rigid motion is stable. This is achieved using only internal actuation. Finally, we show how to derive asymptotically stabilizing feedback laws for relative equilibria where both the internal con guration as well as the rigid motion is unstable. Actuation along the group directions is used in addition to internal actuation. The asymptotic stability is now in a full phase neighborhood of a relative equilibrium, rather than just on a momentum surface.