Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44) that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in Fassò and Sansonetto (2016 J. Nonlinear Sci. 26 519–44). The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an n-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.

On integral principles for nonholonomic systems

Systems of linear fuzzy constraints

In this paper we show that the first integrals of the discrete equation of motion for nonconservative and nonholonomic mechanical systems can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analogue of the generalized theorem of Noether in the Calculus of variations.

The multi-degree-freedom nonholonomic constraint systems were analyzed using method of canonical coordinates. The theorem and characteristic equations of the canonical coordinate’s equations for multi-degree-freedom nonholonomic systems are obtained. The conserved quantities and general solutions of the systems are deduced directly by choosing appropriate canonical coordinates. An example is given to illustrate application of the results.

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

Considers the problem of controlling uncertain nonholonomic mechanical systems performing repetitive motions. It is proposed that a simple and effective solution to this problem can be obtained by first using a reduction procedure to obtain a lower dimensional system which retains the mechanical system structure of the original system, and then controlling the reduced system using a learning algorithm in such a way that the complete system evolves in the desired manner. This approach ensures convergence of the actual system motion to the desired motion despite considerable uncertainty regarding the system dynamic model. The efficacy of the proposed learning controller is illustrated through computer simulations with nonholonomic mechanical systems arising from both explicit kinematic constraints and from symmetries of the system dynamics.

The non-Noether conserved quantity for relativistic nonholonomic controllable mechanical system with variable mass is discussed under special infinitesimal transformations in which the time is not a variable. The differential equations of motion of the system are established. The definition and criterion of the form invariance of the system under special infinitesimal transformations are discussed. The necessary and sufficient conditions under which the form invariance is a Lie symmetry is presented. The conditions under which a non-Noether conserved quantity can be induced by the form invariance and the form of the conserved quantity are obtained. An example is presented to illustrate the application of the result.

The purpose of the present paper is to study the geometry of a sub‐Riemannian manifold and to apply the results to the nonholonomic mechanical systems. First, we construct a linear connection on the horizontal distribution and obtain some Bianchi identities for it. Then, we introduce the horizontal sectional curvature, state a Schur Theorem for sub‐Riemannian geometry, and find a class of sub‐Riemannian manifolds of constant horizontal curvature. Also, we define the horizontal Ricci tensor and scalar curvature, and some sub‐Riemannian differential operators (gradient, divergence, Laplacian), extending some results from Riemannian geometry to the sub‐Riemannian setting. Finally, by using the sub‐Riemannian connection constructed here, we express the Lagrange‐d' Alembert equations for a nonholonomic mechanical system in a form that is similar to the Newton's equations from classical mechanics.

This thesis studies geometric formulations of a slow-fast realization of nonholonomic mechanical control systems mediated by strong friction forces. Such realization is motivated by wheeled mobile robotic systems and viewed as a singular perturbation of nonholonomic dynamics. For sufficiently strong friction forces, the dynamics emit an attractive invariant manifold interpreted as the perturbation of the constraints. This invariant manifold describes small slip velocities at the wheel contact points. We propose a systematic decomposition of the control system into slow and fast directions. The invariant manifold is approximated with a power series, which can be recursively computed. Accordingly, we develop a novel recursive procedure to input-output linearize the slow subsystem by dynamic-state feedback transformations. Using this procedure, design a trajectory-tracking PD controller to compensate for the violations of constraints and closed-loop stability analysis is performed. Our approach is illustrated through a numerical case study.

In this paper, variational techniques are used to analyze the dynamics of nonholonomic mechanical systems with impacts. Implicit nonholonomic smooth Lagrangian and Hamiltonian systems are extended to a nonsmooth context appropriate for collisions. In particular, we provide a variational formulation for implicit nonholonomic mechanical systems with collisions, for those collisions that preserve energy and momentum at the impact. Lastly, the theoretical results are illustrated by examining the example of a rolling disk hitting a wall.

With this paper we recall some of the formal framework developed in classical analytical mechanics with emphasis on constrained systems. It can be used in describing nonholonomic control systems and serve as a starting point for further differential‐geometric investigations.

Recent results in the geometry of constrained systems

This paper outlines some features of general reduction theory as well as the geometry of nonholonomic mechanical systems. In addition to this survey nature, there are some new results. Our previous work on the geometric theory of Lagrangian reduction provides a convenient context that is herein generalized to nonholonomic systems with symmetry. This provides an intrinsic geometric setting for many of the results that were previously understood primarily in coordinates. This solidification and extension of the basic theory should have several interesting consequences, some of which are spelled out in the final section of the paper. Two important references for this work are Cendra, Marsden and Ratiu [2000], hereafter denoted CMR and Bloch, Krishnaprasad, Marsden and Murray [1996], hereafter denoted BKMM.

Nonholonomic redundancy is an intrinsic property of nonholonomic mechanical systems. A free-flying space robot is a nonholonomic mechanical system, and exhibits the presence of nonholonomic redundancy even in the absence of ordinary kinematic redundancy. Like ordinary kinematic redundancy, nonholonomic redundancy can also be utilized while planning trajectories for the system. In the paper, a trajectory planning scheme for a 6-DOF space robot is developed in which nonholonomic redundancy for avoiding joint limits and obstacles is utilized. >