Collaborative Research: Dynamics, Stabilization and Control of Nonholonomic Systems

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Proposal: DMS-0305837PI: Anthony M. Bloch [abloch@math.1sa.umich.edu]Institution: University of Michigan Ann ArborTitle: Collaborative Research: Dynamics, Stabilization and Control of Nonholonomic SystemsABSTRACTThe investigators explore the dynamics of nonholonomic systems, that is, mechanical systems subject to velocity constraints. Proposed research includes studying the existence and properties of momentum conservation laws in nonholonomic dynamics, energy-momentum methods in stability analysis, the qualitative analysis of dynamics of systems with non-free group actions, nonlinear feedback stabilization, and the energy-based approach to tracking problems, both holonomic and nonholonomic. Momentum conservation laws in nonholonomic mechanics are subtle as the classical Noether theorem does not in general apply. This leads to rich dynamics which the investigators study. The role of integrability and discrete symmetries in nonholonomic systems is also analyzed. The investigators study various extensions of the energy-momentum method to both stability, stabilization and control of physical systems. In the nonholonomic setting integrability of the momentum equation is utilized, while in general use is made of the method of controlled Lagrangians.The dynamics of systems with velocity constraints (such as rolling and sliding constraints) has numerous applications in industry; robotics and the dynamics of wheeled vehicles are examples. In applications, stabilization of steady-state motions (such as the straightforward motion of a car at a constant speed) is often desired. The investigators use various tools from the geometric theory of dynamics and control for studying such motions and, more importantly, for programming desired motions in the presence of control forces. The study is ultimately aimed at the design of energy-efficient controllers based on the natural mechanical features of the systems under investigation. The dynamics of single-wheeled vehicles is of special interest because of their exceptional, maneuverability. More complex systems are studied as well.