The Geometric Structure of Nonholonomic 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 ⊂ 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 constraints 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. 1Research partially supported by the DOE contract DEFG0395-ER25251. The dynamics of such a system is described by a set of equations of the following form: g−1ġ = −Anh(r)ṙ + I−1(r)p (1.1) ṗ = ṙH(r)ṙ + ṙK(r)p+ pD(r)p (1.2) M(r)r̈ = δ(r, ṙ, p) + τ. (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 (ṙ, p). The variable τ 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 [BS], 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 [BS]. 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 the group element g, an equation for the nonholonomic momentum p and the reduced Hamilton equations for the shape variables r, pr. The basic theory is illustrated with the snakeboard, as well as a simplified model of the bicycle (see Getz and Marsden [1995]). The latter is an important prototype control system because it is an underactuated balance system. For more details, see Koon and Marsden [1997b]. 1.2 Poisson Geometry On the Hamiltonian side, besides the symplectic point of view, one can also develop the Poisson point of view. Because of the momentum equation, it is natural to let the value of momentum be a variable and for this a Poisson rather than a symplectic viewpoint is more natural. Some of this theory has been started in van der Schaft and Maschke [1994], hereafter denoted [VM]. In our second main result, we build on their work and develop the Poisson reduction for the nonholonomic systems with symmetry. We use this Poisson reduction procedure to obtain specific formulas for the nonholonomic Hamiltonian dynamics. We also show that the equations given by Poisson reduction are equivalent to those given by the Lagrangian reduction via a reduced constrained Legendre transform. Two interesting complications make this effort especially interesting. First of all, as we have mentioned, symmetry need not lead to conservation laws but rather to a momentum equation. Second, the natural Poisson bracket fails to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum that vanishes when the Jacobi identity holds) is an interesting expression involving the curvature of the underlying distribution describing the nonholonomic constraints. We shall explore these in detail in Koon and Marsden [1997c]. Van der Schaft and Maschke [1994] use the Legendre transformation FL : TQ → T ∗Q to define the Hamiltonian H in the standard fashion: H = piq̇ − L, where p = FL(vq) = ∂L/∂q̇, and then write the equations of motion in the Hamiltonian form as ( q̇