Computational Geometric Mechanics and Control of Rigid Bodies.

Abstract

This dissertation studies the dynamics and optimal control of rigid bodies from two complementary perspectives, by providing theoretical analyses that respect the fundamental geometric characteristics of rigid body dynamics and by developing computational algorithms that preserve those geometric features. This dissertation is focused on developing analytical theory and computational algorithms that are intrinsic and applicable to a wide class of multibody systems.

A geometric numerical integrator, referred to as a Lie group variational integrator, is developed for rigid body dynamics. Discrete-time Lagrangian and Hamiltonian mechanics and Lie group methods are unified to obtain a systematic method for constructing numerical integrators that preserve the geometric properties of the dynamics as well as the structure of a Lie group. It is shown that Lie group variational integrators have substantial computational advantages over integrators that preserve either one of none of these properties. This approach is also extended to mechanical systems evolving on the product of two-spheres.

A computational geometric approach is developed for optimal control of rigid bodies on a Lie group. An optimal control problem is discretized at the problem formulation stage by using a Lie group variational integrator, and discrete-time necessary conditions for optimality are derived using the calculus of variations. The discrete-time necessary conditions inherit the desirable computational properties of the Lie group variational integrator, as they are derived from a symplectic discrete flow. They do not exhibit the numerical dissipation introduced by conventional numerical integration schemes, and consequently, we can efficiently obtain optimal controls that respect the geometric features of the optimality conditions.

The approach that combines computational geometric mechanics and optimal control is illustrated by various examples of rigid body dynamics, which include a rigid body pendulum on a cart, pure bending of an elastic rod, and two rigid bodies connected by a ball joint. Since all of the analytical and computational results developed in this dissertation are coordinate-free, they are independent of a specific choice of local coordinates, and they completely avoid any singularity, ambiguity, and complexity associated with local coordinates. This provides insight into the global dynamics of rigid bodies.