Nonholonomic Geometry, Mechanics and Control

Abstract

This dissertation is concerned with dynamic modeling and kinematic control of constrained mechanical systems with symmetry from a geometric point of view. Constraints are defined via the characteristics of distributions or codistributions on the tangent bundle (velocity phase space) of configuration space. Lie symmetry groups acting on the systems are assumed to leave both Lagrangian and constraints invariant. As a special case of mechanical systems with holonomic constraints, we rigorously analyze the kinematics and dynamics of floating, planar four-bar linkages. The analyses include topological description of the configuration space, symplectic and Poisson reductions of the dynamics and bifurcation of relative equilibria. for kinematic control of nonholonomic systems, we mainly study the related optimal control problem for a system consisting of a rigid body with two oscillators. In particular, the intrinsic formulation and explicit solvability of necessary conditions for the optimal control are investigated from a Hamiltonian point of view. In the study of the dynamics of Lagrangian systems with constraints, the nonholonomic distributions are defined via arbitrary choices of principal connections. We show that, under our hypotheses on constraints and exterior force, the dynamics of a nonholonomic Lagrangian system with non-Abelian symmetry can be reduced to a lower dimensional space determined by the principal fiber bundle. The reduced dynamic equations are formulated explicitly. This formulation generalizes the one for classical Chaplygin systems which possess Abelian symmetry, and the one having non-Abelian symmetry but with linear constraints. In addition, if a special principal connection, that is, the mechanical connection by Kummer and Smale, is considered, our formulation for nonholonomic systems also leads to the one in Lagrangian reduction discovered recently by Marsden and Scheurle. The results of this dissertation have direct application in space robotics and nonholonomic motion planning in robotics.