Dmitry V. Zenkov

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.

Discrete nonholonomic LL systems on Lie groups

This paper studies discrete nonholonomic mechanical systems whose configuration space is a Lie group G. Assuming that the discrete Lagrangian and constraints are left-invariant, the discrete Euler–Lagrange equations are reduced to the discrete Euler–Poincare–Suslov equations. The dynamics associated with the discrete Euler–Poincare–Suslov equations is shown to evolve on a subvariety of the Lie group G. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and discrete momentum conservation is discussed.

Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization

Abstract We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton–Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton–Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.

Dynamics of the n-dimensional Suslov problem

Abstract In this paper we study the dynamics of a constrained generalized rigid body (the Suslov problem) on its full phase space. We use reconstruction theory to analyze the qualitative dynamics of the system and discuss differences with the free rigid body motion. Use is made of the so-called quasi-periodic Floquet theory.

Invariant measures of nonholonomic flows with internal degrees of freedom

In this paper we study measure preserving flows associated with nonholonomic systems with internal degrees of freedom. Our approach reveals geometric reasons for the existence of measures in the form of an integral invariant with smooth density that depends on the internal configuration of the system. Mathematics Subject Classification: 37J15, 37J60, 70F25

The geometry of the Routh problem

SummaryIn this paper the motion without sliding of a homogeneous ball on a surface of revolution is studied. It is shown that the necessary condition for stability of stationary periodic motions of the ball obtained by Routh is also a sufficient one. We prove that the nondegenerate invariant manifolds are diffeomorphic to unions of invariant tori filled with quasiperiodic motions.

The Lyapunov-Malkin theorem and stabilization of the unicycle with rider

This paper analyzes stabilization of a nonholonomic system consisting of a unicycle with rider. It is shown that one can achieve stability of slow steady vertical motions by imposing a feedback control force on the rider’s limb.

Rotation invariant topology coding of 2D and 3D objects using Morse theory

In this paper, we propose a numerical algorithm for extracting the topology of a three-dimensional object (2 dimensional surface) embedded in a three-dimensional space /spl Ropf//sup 3/. The method is based on capturing the topology of a modified Reeb graph by tracking the critical points of a distance function. As such, the approach employs Morse theory in the study of translation, rotation, and scale invariant skeletal graphs. The latter are useful in the representation and classification of objects in /spl Ropf//sup 3/.

Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System

Matching techniques are developed for discrete mechanical systems with symmetry. We describe new phenomena that arise in the controlled Lagrangian approach for mechanical systems in the discrete context. In particular, one needs to either make an appropriate selection of momentum levels or introduce a new parameter into the controlled Lagrangian to complete the matching procedure. We also discuss digital and model predictive control.

Matching and stabilization of low-dimensional nonholonomic systems

We show how a generalized matching technique for stabilization may be applied to the Routhian associated with a low-dimensional nonholonomic system. The theory is illustrated with a simple model-a unicycle with rider.