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.