### Hamiltonian and Lagrangian Approaches to Nonholonomic System

Many problems in robotics, dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Despite considerable advances in both Hamiltonian and Lagrangian sides of the theory, there remains much to be done, and this study makes contributions in three important areas.

First, we establish necessary conditions for optimal control using the ideas of Lagrangian reduction. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. Lagrangian reduction can do in one step what one can alternatively do by applying Pontryagin Maximum Principle followed by Poisson reduction. We apply the techniques to some known examples of optimal control on Lie groups and principal bundles. More importantly, we extend the method to the case of nonholonomic systems with a nontrivial momentum equation, such as the snakeboard.

Second, we compare the Hamiltonian (symplectic) approach to nonholonomic systems with Lagrangian approach. There are many differences between these approaches, and it was not obvious how they were equivalent. For example, Bloch, Krishnaprasad, Marsden and Murray [1996] developed the momentum equation, the reconstruction equation and the reduced Lagrange-d'Alembert equations, which are important for control applications, and it is not obvious how these correspond to the developments in Bates and Sniatycki [1993]. Our second result establishes specific links between these two sides and uses the ideas and results of each to shed light on the other, deepening our understanding of both approaches. We treat a simplified model of the bicycle and obtain new and interesting results.

We also develop the Poisson point of view for nonholonomic systems. Some of this theory has been started in van der Schaft and Maschke [1994]. In our third result, we develop the Poisson reduction for nonholonomic systems with symmetry, which enables us to obtain specific formulas for the Hamiltonian dynamics. Moreover, we show that the equations given by the Poisson reduction are equivalent to those given by the Lagrangian reduction.

We hope that these results will help lay a firm foundation for further developments of control, stability and bifurcation theories for such systems.

PAPERS
• The Poisson Reduction of NonholonomicMechanical Systems, Reports on Mathematical Physics, 42, 101-134. (Koon, W. S. and J. E. Marsden [1998], pdf)

• The Geometric Structure of Nonholonomic Mechanics, Proc. IEEE CDC, 36, 4856-4862. (Koon, W. S. and J. E. Marsden [1998], pdf)

• The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems, Reports on Mathematical Physics, 40, 21-62. (Koon, W. S. and J. E. Marsden [1997], pdf)

• Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction, SIAM Journal on Control and Optimization, 35, 901-929. (Koon, W.S. and J.E. Marsden [1997], pdf)

THESIS
• Koon, W.S. [1997], Reduction, Reconstruction and Optimal Control for Nonholonomic Mechanical Systems with Symmetry, Dedicated to My Parents, Soo Wu Chow and Ziang Yuen Kuen. (pdf)

PRESENTATIONS

• The Poisson Reduction of Nonholonomic Mechanical Systems, AFSOR/Caltech Workshop on Mechanics, Dynamics and Control, December 1997, California Institute of Technology.

• The Geometric Structure of Nonholonomic Mechanics, 36th IEEE Conference on Decision and Control, December 1997, San Diego, California.

• The Poisson Reduction of Nonholonomic Mechanical Systems, Workshop on Nonholonomic Constraints in Dynamics, August 1997, University of Calgary, Calgary, Canada.

• Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry, AMS 1997 Summer Research Institute on Differential Geometry and Contro, University of Colorado, Boulder, Colorado.

• The Poisson Reduction of Nonholonomic Mechanical Systems, Fourth SIAM Conference on Application of Dynamcial Systems, May 1997, Snowbird, Utah, USA.

• The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems, seminar at Dept. of Mechanical Engineering and Applied Mechanics, April 1997, University of Pennsylvania.

• Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry, Control and Dynamical Systems Seminar, October 1995, California ofInstitute of Technology.