The Geometric Structure of Nonholonsmic Mechanics

Many important problems in multibody dynamics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory. Despite considerable advances on both Hamiltonian and Lagrangian sides of the theory, there remains much to do. We report on progress on two of these fronts. The first is a Poisson description of the equations that is equivalent to those given by Lagrangian reduction, and second, a deeper understanding of holonomy for such systems. These results promise to lead to further progress on the stability issues and on locomotion generation. 1 Symplectic and Poisson Geometry of Nonholonomic Systems Bloch, Krishnaprasad, Marsden and Murray [1996], hereafter denoted [BKMM], applied methods of geometric mechanics to the Lagrange-d’Alembert formulation and extended the use of connections and momentum maps associated with a given symmetry group to this case. The resulting framework, including the nonholonomic momentum and nonholonomic mechanical connection, provides a setting for studying nonholonomic mechanical control systems that may have a nontrivial evolution of their nonholonomic momentum. The setting is a configuration space Q with a (nonintegrable) distribution D c TQ describing the constraints. For simplicity, we consider only homogeneous velocity constraints. We are given a Lagrangian L on TQ and a Lie group G acting on the configuration space that leaves the constraint,s and the Lagrangian invariant. In many example, the group encodes position and orientation information. For example, for the snakeboard, the group is SE(2) of rotations and translations in the plane. The quotient space Q/G is called shape space. lResearch partially supported by the DOE contract DEFG0395-ER2525 1. Jerrold E. Marsden Control and Dynamical Systems California Institute of Technology 116-81 Pasadena, CA 91125 [email protected] The dynamics of such a system is described by a set of equations of the following form: g-lg = -dnh(r)7: + I l ( r ) p (1.1) f i = T H ( r ) i t i T ~ ( r ) p + p T ~ ( r ) p (1.2) M(r)j: = G ( r , t , p ) +T. (1.3) The first equation is a reconstruction equation for a group element g, the second is an equation for the nonholonomic momentum p (not conserved in general), and the third are equations of motion for the reduced variables r which describe the “shape” of the system. The momentum equation is bilinear in ( + , p ) . The variable 7 represents the external forces applied to the system, and is assumed to affect only the shape variables, i.e., the external forces are G-invariant. Note that the evolution of the momentum p and the shape r decouple from the group variables. This framework has been very useful for studying controllability, gait selection and locomotion for systems such as the snakeboard. It has also helped in the study of optimality of certain gaits, by using optimal control ideas in the context of nonholonomic mechanics (Koon and Marsden [1997a] and Ostrowski, Desai and Kumar [1997]). Hence, it is natural to explore ways for developing similar procedures on the Hamiltonian side. 1.1 Symplectic Reduction Bates and Sniatycki [1993], hereafter denoted [BSI, developed the symplectic geometry on the Hamiltonian side of nonholonomic systems, while [BKMM] explored the Lagrangian side. It was not obvious how these two approaches were equivalent, especially how the momentum equation, the reduced Lagrange-d’ Alembert equations and the reconstruction equation correspond to the developments in [BSI. Our first main result establishes the 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 points of view. For example, in proving the equivalence of the Lagrangian reduction and the symplectic reduction, we have shown where the momentum equation is lurking on the Hamiltonian side and how this is related to the organization of the dynamics of nonholonomic systems with symmetry into the three parts displayed above: a reconstruction equation for 0-7803-3970-8/97 $10.00