Nonholonomic Mechanical Systems Research Articles

Nonholonomic mechanics: A practical application of the geometrical theory on fibred manifolds to a planimeter motion

A geometrical theory of general nonholonomic mechanical systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was developed in 1990s by Krupková. 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 semiholonomic, i.e. integrable. On the other hand, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. In this paper we study the planimeter—a mechanism for measuring areas which belongs to mechanical systems subjected to constraint conditions containing among others a true nonholonomic one.We study the planimeter motion using the above mentioned Krupková's approach. The results of numerical solutions of constrained equations of motion, derived within the theory, are in a good agreement with theoretical ones and thus they confirm the possibility of direct application of the theory to practical situations.

Modeling and robust backstepping control of an underactuated quadruped robot in bounding motion

SUMMARYIn this paper, a model-based exponential stabilization of a quadruped robot is studied in bounding motion. The dynamics of the five-link planar underactuated mechanical model of the quadruped robot with four actuated joints system is derived. It is shown that the dynamical equation of the proposed simplified model belongs to a class of second-order nonholonomic mechanical systems which cannot be stabilized by any smooth time-invariant state feedback. Utilizing a coordinate transformation based on the so-called normalized momentum, a robust backstepping control method is presented for the quadruped robot. Both theoretical analysis and numerical simulations show that the robust backstepping controller can stabilize the underactuated quadruped robot so that it could balance on its rear legs and track a desired trajectory. Despite the model parameter uncertainties, the robustness of the controller is maintained. The simulation results show the effectiveness of the proposed method.

Design of a robust neural network-based tracking controller for a class of electrically driven nonholonomic mechanical systems

This paper addresses the problem of designing robust tracking controls for a class of uncertain nonholonomic systems actuated by brushed direct current (DC) motors. This class of electrically driven nonholonomic mechanical systems can be perturbed by plant uncertainties, unmodeled time-varying perturbations, and external disturbances. An adaptive neural network-based dynamic feedback tracking controller will be developed such that all the states and signals of the closed-loop system are bounded and the trajectory tracking error is as small as possible. Consequently, for practical applications, the intelligent robust tracking control scheme developed here can be employed to handle a broader class of electrically driven nonholonomic systems in the presence of high-degree time-varying uncertainties. Finally, simulation examples are presented to demonstrate the effectiveness of the proposed control algorithms.

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton–Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.

Brachistochronic motion of nonholonomic nonlinear mechanical system of the Chaplygin type

In this paper, the differential equations of brachistochronic motion of nonholonomic nonlinear system of the Chaplygin type were determined. The additional forces Rα realizing the corresponding motion of the system together with the given forces were also determined. Lastly, the brachistochrone was solved for a concrete model using mathematical numerical methods.

Exploring the analogy for adaptive attitude control of a ground-based satellite system

Equations of motion for a special system pertaining to the class of mixed nonholonomic mechanical systems are studied in the modern setting of geometric mechanics. The presented attitude control test bed is intended to provide an experimental facility that, in certain senses, emulates the dynamics of on-orbit conditions in the laboratory site, allowing the evaluation of path planning and feedback control algorithms. This paper demonstrates the feasibility of the approach and proposes a concurrent solution to the attitude tracking control problem that, due to uncertainties of the parameters, is likely to require effective adaptive aptitudes. Moreover, the invariance with respect to Lie group actions of governing dynamics and measurable output readings has allowed the investigation of controllability and observability in an intrinsic manner.

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

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.

Simulation of Nonholonomic Mechanical Systems Using Algorithmic Differentiation

AbstractThe purpose of this contribution is to introduce a new approach for simulating mechanical systems with nonholonomic constraints. The technique of algorithmic differentiation is employed together with Hamel’s equations of motion permitting to incorporate the nonholonomic constraints directly instead of generating a system of differential-algebraic equations. To apply the proposed simulation approach, the user only needs to state the kinetic and potential energies, the nonconservative forces, and the nonholonomic constraints of the system to be simulated. The procedure is demonstrated by an illustrating example of a nonholonomic moving robot.

Variational Integrators for Hamiltonizable Nonholonomic Systems

We report on new applications of the Poincare and Sundman time-transformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these non-variational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results.

The Borkhoffian expression of Boltzmann-Hamel equation of nonholonomic system and its generalized symplectic geometric algorithm

For a Boltzmann-Hamel equation of nonholonomic mechanical system, when it meets certain conditions, the Boltzmann-Hamel equation can be transformed into a Birkhoffian system. By constructing the generating function, the system is investigated numerically using the generalized symplectic geometric algorithm of the nonautonomous Birkhoffian system. Compared with the above-mentioned algorithm with the classical Runge-Kutta method, Birkhoffian symplectic scheme is very accurate in a long-term tracing.

Conformal invariance and conserved quantity of Mei symmetry for higher-order nonholonomic system

The conformal invariance and conserved quantity of Mei symmetry for a higher-order nonholonomic mechanical system are presented. Introducing an infinitesimal transformation group and infinitesimal generator vector, the definition of conformal invariance of Mei symmetry and the determining equation for the holonomic system which corresponds to a higher-order nonholonomic system are provided, and the relationship between Mei symmetry and conformal invariance of the system is discussed. The basis of restriction equations and additional restriction equations, the conformal invariance of weak and strong Mei symmetry for the higher-order nonholonomic mechanical system is constructed. With the aid of a structure equation that the gauge function satisfies, the system’s corresponding conserved quantity is derived. Finally, an example is given to illustrate the application of the method and its result.

The non-holonomic double pendulum: An example of non-linear non-holonomic system

An example of physically realizable non-linear non-holonomic mechanical system is proposed. The dynamical equations are written following a general method proposed in an earlier paper. In order to make this paper self-contained, an improved and shortened approach to the dynamics of non-holonomic systems is illustrated in preliminary sections.

Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system

Perturbation to Noether symmetries and adiabatic invariants of discrete nonholonomic nonconservative mechanical systems on an uniform lattice are investigated. Firstly, we review Noether symmetry and conservation laws of a nonholonomic nonconservative system. Secondly, we study continuous Noether symmetry of a discrete nonholonomic system, give the Noether symmetry criterion and theorem of discrete corresponding holonomic system and nonholonomic system. Thirdly, we study perturbation to Noether symmetry of the discrete nonholonomic nonconservative system, give the criterion of perturbation to Noether symmetry for this system, and based on the definition of adiabatic invariants, we construct the theorem under which can lead to Noether adiabatic invariants for this system, and the forms of discrete Noether adiabatic invariants are given. Finally, we give an example to illustrate our results.

A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (II)

We show that the sub-Riemannian connection ∇, constructed in the first part of our study [2], is the one which enables us to 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 (cf. (2.21)). Also, we define the generalized Chaplygin systems and show that the Lagrange–d'Alembert equations for these systems depend only on some horizontal variables. Two examples are given to illustrate our theory in comparison with other studies.

Lie-Mei symmetry and conserved quantities of the Rosenberg problem

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems. The Lie—Mei symmetry and the conserved quantities of the Rosenberg problem are studied. For the Rosenberg problem, the Lie and the Mei symmetries for the equation are obtained, the conserved quantities are deduced from them and then the definition and the criterion for the Lie—Mei symmetry of the Rosenberg problem are derived. Finally, the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie—Mei symmetry.

Jacobi Last Multiplier Method for Equations of Motion of Constrained Mechanical Systems

The Jacobi last multiplier method for holonomic and nonholonomic mechanical systems is studied and some examples are given to attempt applications of the method.

A Field Integration Method for a Nonholonomic Mechanical System of Non-Chetaev's Type

A field integration method for a nonholonomic mechanical system of non-Chetaev's type is studied. The differential equations of the motion of the system are established. The solution of the corresponding holonomic system for the nonholonomic system is obtained by the field method. The restriction of nonholonomic constrained to initial conditions is added and the solution of the nonholonomic mechanical system of non-Chetaev's type is provided. An example is presented to illustrate the application of the results.

Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.

Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems**Project supported by the National Natural Science Foundation of China (Grant No. 10372053).

This paper presents the Mei symmetries and new types of non-Noether conserved quantities for a higher-order nonholonomic constraint mechanical system. On the basis of the form invariance of differential equations of motion for dynamical functions under general infinitesimal transformation, the determining equations, the constraint restriction equations and the additional restriction equations of Mei symmetries of the system are constructed. The criterions of Mei symmetries, weak Mei symmetries and strong Mei symmetries of the system are given. New types of conserved quantities, i.e. the Mei symmetrical conserved quantities, the weak Mei symmetrical conserved quantities and the strong Mei symmetrical conserved quantities of a higher-order nonholonomic system, are obtained. Then, a deduction of the first-order nonholonomic system is discussed. Finally, two examples are given to illustrate the application of the method and then the results.

Nonholonomic Hamilton–Jacobi theory via Chaplygin Hamiltonization

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.