Implicit Nonholonomic Mechanics with Collisions

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.

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.

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.

Nonholonomic systems provide an important class of mechanical control systems. One reason for this importance is that nonintegrability is essential to both the mechanics and the control: Nonintegrable constraint distributions are the essence of nonholonomic systems, while a nonintegrable distribution of control vector fields is the key to controllability of nonlinear systems. We will learn how these two different types of nonintegrability work together when we study control of nonholonomic mechanical systems.

ABSTRACTSegway is a self-balancing motorized two-wheeled vehicle which is able to carry the human body. In the presented paper, a problem of nonholonomic constrained mechanical systems is treated. New methods in nonholonomic mechanics are applied to a problem of a two-wheeled self-balancing robots motion ‘SEGWAY’. This method of the geometrical theory of general nonholonomic constrained systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupková and others. The equations of motion of a two-wheeled self-balancing robot are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupková approach. Additionally, the results of numerical solutions of constrained equations of motion derived within the theory are in good agreement with results of (1) [Maddahi, A., Shamekhi, A. H., & Ghaffari, A. (2015). A Lyapunov controller for self-balancing two-wheeled vehicles. Robotica, 33(1), 225–239]. using Lyapunov's feedback control design technique. The existence, continuity, and uniqueness of the solution for the proposed control system are proved utilizing the Filippov's solution (2). And with fuzzy controller proposed by [Qian, Q., Wu, J., & Wang, Z. (2017). A novel configuration of two-wheeled self-balancing robot/Nova konfiguracija samobalansirajuceg robota na dva kotaca (Original scientific paper/Izvorni znastveni clanak). Tehnicki Vjesnik-Technical Gazette, 24(2), 459–465].

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.

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.

The theory of non-holonomic mechanical systems appeared when it was seen that the classical Lagrangian formalism (corresponding to the holonomic mechanical systems) cannot be applied in case of some very simple problems (e.g., the rolling without sliding of a rigid solid on a fixed plane). After that Lindelöf’s error in this direction has been detected by Chaplygin, one has obtained many formulations and there have been made many studies by Appell, Bobylev, Chaplygin, Tsenov, Hamel, Hertz, Maggi, Voronets, Zhukovskiĭ and others; the actual research in this direction is very rich.KeywordsMechanical SystemConstraint RelationFixed PlaneAcceleration EnergyMovable FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mechanical systems have traditionally provided a fertile area of study for researchers interested in nonlinear control, due to the inherent nonlinearities and the Lagrangian structure of these systems. Recently, a great deal of emphasis has been placed on studying systems with nonholonomic constraints, including mobile wheeled robots and multiple-trailer vehicles, where the wheels provide a no-slip velocity constraint. In this paper, a new methods in non-holonomic mechanics are applied to a problem of an ackerman steered car motion for the first time. This method of the geometrical theory of general nonholonomic constrained systems on fibered manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupkov´a in 1990’s. The relevance of this theory for general types of nonholonomic constraints, not only linear or affine ones, was then verified on appropriate models. Frequently considered constraints on real physical systems are based on rolling without sliding, i.e. they are holonomic, or semi-holonomic, i.e. integrable. Moreover, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. On the other hand, the equations of motion of an ackerman steered car are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupkov´a approach. The results of numerical solutions of constrained equations of motion, derived within the theory, are presented and thus they open the possibility of direct application of the theory to practical situations in engineers.

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.

This paper enlarges the reservoir of solved tutor problems in non-holonomic mechanics at the undergraduate level of physics education. Unlike other, rather artificial, solved problems typically used, the streetboard-rider locomotion problem presented here represents an appealing contemporary real-world problem with interesting applications in a popular sport and in robotics. In this paper, the streetboard motion is discussed from the physical point of view. We show that the interesting snake-like motion performed by streetboard riders stems from its non-holonomic nature. The related non-holonomic constraints are derived and the problem of the mechanical system subjected to these non-holonomic constraints is solved using methods appropriate to the undergraduate university level. The analytical solution for the circular motion and the numerical solution for the general motion are obtained, the physical meaning of the derived constrained forces is investigated and the results are discussed with respect to observed streetboard-rider locomotion. The brief outline of the paper can be used as a demonstration example in non-holonomic mechanics lessons, while the paper itself establishes an original undergraduate computational student project in theoretical mechanics.

The aim of this study is to present an alternative way to deduce the equations of motion of general (i. e. also nonlinear) nonholonomic constrained systems starting from the d’Alembert principle and proceeding by an algebraic procedure. The two classical approaches in nonholonomic mechanics – Cˇ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\check{\ extrm{C}}}$$\\end{document}etaev method and vakonomic method – are treated on equal terms, avoiding integrations or other steps outside algebraic operations. In the second part of the work, we compare our results with the standard forms of the equations of motion associated with the two method and we discuss the role of the transpositional relation and of the commutation rule within the question of equivalence and compatibility of the Cˇ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\check{\ extrm{C}}}$$\\end{document}etaev and vakonomic methods for general nonholonomic systems.

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 equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, that unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.