Geometry, Dynamics and Control of Coupled Systems

Abstract

In this dissertation, we study the dynamics and control of coupled mechanical systems. A key feature of this work is the systematic use of modern geometric mechanics, including methods based on symplectic geometry, Lie symmetry groups, reductions, lagrangian mechanics and hamiltonian mechanics to investigate specific Eulerian manybody problems. A general framework for gyroscopic systems with symmetry is introduced and analyzed. The influence of the gyroscopic term (linear term in Lagrangian) on the dynamical behavior is exploited. The notion of gyroscopic control is proposed to emphasize the role of the gyroscopic term in designing control algorithms. The block-diagonalization techniques associated to the energy-momentum method which proved to be very useful in determining stability for simple mechanical systems with symmetry are successfully extended to gyroscopic systems with symmetry. The techniques developed here are applied to several interesting mechanical systems. These examples include the dual-spin method of attitude control for artificial satellites, a multi-body analog of the dual-spin problem a rigid body with momentum wheels in a central gravitational force field, and a rigid body with momentum wheels and a flexible attachment.