Simulating Nonholonomic Dynamics

This paper develops different discretization schemes for nonholonomic mechanical systems through a discrete geometric approach. The proposed methods are designed to account for the special geometric structure of the nonholonomic motion. Two different families of nonholonomic integrators are developed and examined numerically: the geometric nonholonomic integrator (GNI) and the reduced d'Alembert-Pontryagin integrator (RDP). As a result, the paper provides a general tool for engineering applications, i.e. for automatic derivation of numerically accurate and stable dynamics integration schemes applicable to a variety of robotic vehicle models

Conservation of energy and momenta in nonholonomic systems with affine constraints

We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their `gauge-like' generalizations, in time-independent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the so-called reaction-annihilator distribution

Symmetry Reduction of Optimal Control Systems and Principal Connections

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure

Poisson reduction for nonholonomic mechanical systems with symmetry

This paper continues the work of Koon and Marsden [1997b] that began the comparison of the Hamiltonian and Lagrangian formulations of nonholonomic systems. Because of the necessary replacement of conservation laws with the momentum equation, it is natural to let the value of momentum be a variable and for this reason it is natural to take a Poisson viewpoint. Some of this theory has been started in van der Schaft and Maschke [1994]. We build on their work, further develop the theory of nonholonomic Poisson reduction, and tie this theory to other work in the area. We use this reduction procedure to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic momentum equation and the reduced Lagrange d’Alembert equations in Hamiltonian form. We also show that these equations are equivalent to those given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden and Murray [1996]. Because of the results of Koon and Marsden [1997b], this is also equivalent to the results of Bates and Sniatycki [1993], obtained by nonholonomic symplectic reduction. 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), or equivalently, the Schouten bracket, is an interesting expression involving the curvature of the underlying distribution describing the nonholonomic constraints. The Poisson reduction results in this paper are important for the future development of the stability theory for nonholonomic mechanical systems with symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they should be useful for the development of the powerful block diagonalization properties of the energy-momentum method developed by Simo, Lewis and Marsden [1991]

The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.Comment: 25 page

The energy–momentum method for the stability of non-holonomic systems

In this paper we analyze the stability of relative equilibria of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the the rolling disk, the roller racer, and the rattleback top