Aspects of Geometric Mechanics and Control of Mechanical Systems

Abstract

Many interesting control systems are mechanical control systems. In spite of this, there has not been much effort to develop methods which use the special structure of mechanical systems to obtain analysis tools which are suitable for these systems. In this thesis we take the first steps towards a methodical treatment of mechanical control systems.

First we begin to develop a framework for analysis of certain classes of mechanical control systems. In the Lagrangian formulation we study "simple mechanical control systems" whose Lagrangian is "kinetic energy minus potential energy." We propose a new and useful definition of controllability for these systems and obtain a computable set of conditions for this new version of controllability. We also obtain decompositions of simple mechanical systems in the case when they are not controllable. In the Hamiltonian formulation we study systems whose control vector fields are Hamiltonian. We obtain decompositions which describe the controllable and uncontrollable dynamics. In each case, the dynamics are shown to be Hamiltonian in a suitably general sense.

Next we develop intrinsic descriptions of Lagrangian and Hamiltonian mechanics in the presence of external inputs. This development is a first step towards a control theory for general Lagrangian and Hamiltonian control systems. We also study systems with constraints. We first give a thorough overview of variational methods including a comparison of the "nonholonomic" and "vakonomic" methods. We also give a generalised definition for a constraint and, with this more general definition, we are able to give some preliminary controllability results for constrained systems.