Nonholonomic Mechanical Systems Research Articles

Connection of first integrals with particular solutions of the nonsimultaneous variational equations for nonholonomic systems

In this paper, the nonsimultaneous variational equations of the nonholonomic mechanical systems are presented, and their solutions are studied. It is proven that, under some conditions, a particular solution of the nonsimultaneous variational equations can be obtained by using a first integral. At the end of the paper, an example is given to illustrate the applications of the results.

Form invariance and Lie symmetry of variable mass nonholonomic mechanical system

The form invariance and Lie symmetry of a variable mass nonholonomic mechanical system is studied. The definition and the criterion and the conserved quantity of form invariance and Lie symmetry for the variable mass nonholonomic mechanical system are given. The relation between the form invariance and Lie symmetry is obtained. An example is given to illustrate the application of the result.

First integrals of the discrete nonconservative and nonholonomic systems

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.

A Lie algebroid framework for non-holonomic systems

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

ステアリングを持つ3叉移動機構の制御

This paper introduces and describes a new type of wheeled locomotor which we refer to as a “trident steering walker.” This wheeled locomotor is a nonholonomic mechanical system, and it consists of an equilateral triangle base, three rotational joints, three links, and four steering systems. The equilateral triangle base has a steering system at its center of mass. Each apex of the base has a rotational joint coupling the base and a link which has a steering system at its middle point. The wheeled locomotor transforms the rotations of the three joints into its movement by using the four steering systems. This means that the wheeled locomotor makes undulatory locomotion in which it transforms the change of its internal shape into the generation of its net displacement. Especially, we assume a virtual joint at the tip of the first link. The virtual joint couples the first link and a virtual link which has a virtual axle at its middle point and which has a virtual steering system at its tip. We prove that, by assuming such virtual mechanical elements, it is possible to convert the kinematical equation of the trident steering walker into five-chain, single-generator chained form in a mathematical framework, differential geometry. Based on chained form, we derive a path following feedback control method which makes the trident steering walker follow a straight line. The validity of the mechanical design of the trident steering walker, the conversion of its kinematical equation into chained form, and the path following feedback control method has been verified by computer simulations.

424 ラグランジュ力学の幾何学的方法について : 非ホロノミック系の定式化

424 ラグランジュ力学の幾何学的方法について : 非ホロノミック系の定式化

Form invariance of Raitzin's canonical equations of a nonholonomic mechanical system

A form invariance of Raitzin's canonical equations of a nonholonomic mechanical system is studied. The Raitzin canonical equations of the system are established. The definition and criterion of the form invariance in the system under infinitesimal transformations of groups are given. The relation between the form invariance and the conserved quantity of the system is obtained and an example is also given to illustrate the application of the result.

A Unified Approach to the Energy‐Consistent Numer‐ical Integration of Nonholonomic Mechanical Systems and Flexible Multibody Dynamics

AbstractThe present work deals with the energy‐consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints. The underlying differentialalgebraic equations (DAEs) with index three are directly discretized. This approach makes possible the development of a new energy‐consistent time‐stepping scheme for general nonholonomic systems. In particular, both nonholonomic problems from rigid body dynamics as well as flexible multibody dynamics can be treated in a unified manner. In this connection specific constrained formulations of rigid bodies and geometrically exact beams are presented. Moreover, the newly developed discrete null space method is applied to achieve a size‐reduction and an improved conditioning of the discrete system. The numerical examples deal with a nonholonomic rigid body system and a flexible multibody system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A note on the conservation of energy and volume in the setting of nonholonomic mechanical systems

This note concerns the analysis of conservation of energy and volume for a series of well known examples of nonholonomic mechanical systems, with linear and non-linear constraints, and aims to make evident some geometric aspects related with them.

Robust Motion/Force Control of Uncertain Holonomic/ Nonholonomic Mechanical Systems

In this paper, robust control strategies are presented systematically for both holonomic mechanical systems and a large class of nonholonomic mechanical systems in the presence of uncertainties and disturbances. First, robust control strategies are presented for both kinds of systems using the bounds of system parameters, respectively. Then, adaptive robust control strategies are presented by tuning the parameter estimates online. Proportional plus integral feedback control is used for force control for the benefit of real-time implementation. The proposed control strategies guarantee that the system motion converges to the desired manifold with prescribed performance while the constraint force remains bounded.

A non-Noether conserved quantity, i.e. Hojman conserved quantity, for nonholonomic mechanical systems

A non-Noether conserved quantity, i.e. Hojman conserved quantity, of the nonholonomic mechanical systems is presented. Using the special Lie symmetry under infinitesimal transformations in which the time is not variable, the determining equations, the constrained restriction equations and the additional restriction equations of the nonholonomic mechanical systems are given. The Hojman conserved quantity of the corresponding holonomic systems, the weakly Hojman conserved quantity and the strongly Hojman conserved quantity of the nonholonomic systems are obtained. An example is given to illustrate the application of the result.

Form invariance and Hojman conserved quantity for nonholonomic mechanical systems

A non-Noether conserved quantity, i.e. Hojman conserved quantity，constructed by using a form invariance for the nonholonomic mechanical systems is presented. Under special infinitesimal transformations in which the time is not changed, the determining equations of the special form invariance, the constrained restriction equations, the additional restriction equations, and the definitions of the weak form invariance and the strong form invariance of the nonholonomic mechanical systems are given. The condition under which the special form invariance is a special Lie symmetry are obtained. From the special form invariance, the Hojman conserved quantity of the corresponding holonomic systems, the weak Hojman conserved quantity and the strong Hojman conserved quantity of the nonholonomic systems are obtained. And two examples are given to illustrate the application of the result.

The theory of the generalized magnus effect for non-holonomic mechanical systems

The motion of a carriage with two wheeled pairs over a rough horizontal plane is investigated in the following cases: (1) inertial motion, (2) when there is an elastic constraint which produces a restoring moment of the forces when the axis of rotation of the leading wheeled pair deviates from the unperturbed position, and (3) when there is a small harmonic moment between the leading wheeled pair and the platform. The properties of the exact solution of the system is analysed in the first case. In the second, using the method of averaging, it is shown that for small oscillations of the leading wheeled pair with respect to the platform, after a transition process, motion of the centre of mass of the system with constant velocity, proportional to the initial amplitude of the oscillation, occurs. In the third case, the average motion of the centre of mass occurs with a constant acceleration, the value of which is estimated using the asymptotic multiscale method.

The controllability of non-holonomic mechanical systems with constrained controls

A cycle of investigations, related to the problem of the controllability of non-linear dynamical systems, is developed. Systems which have a mechanical nature (wheeled means of transportation, transport and machining systems, and manipulators) are considered. Controllability criteria are established for the general class of mechanical systems which may contain non-holonomic constraints; similar criteria were established previously for holonomic systems in [1–4]. The controllability conditions obtained have a clear physical meaning. For example, for the controllability of a manipulation robot it is necessary that the control forces should predominate over the other generalized forces (the weight forces, and the resistance forces due to the external medium), Predominance is necessary as regards the amplitude of the forces. Additional conditions are related to the properties of constraints imposed on the system. This essentially requires that the constraint relations allow of the possibility of an appropriate change in the coordinates and velocities of the mechanical system in the region investigated.

The stability and stabilization of the steady motions of a class of non-holonomic mechanical systems

The stability of the steady motions and the controllability of a class of non-holonomic mechanical systems under the action of potential and control forces are investigated. A problem of the stability of the steady motion of a three-wheeled vehicle, taking into account the inertia of the wheels, which is an example of systems of this class, is considered.

The stability of the steady motions of non-holonomic mechanical systems with cyclic coordinates

The stability of the steady motions of non-holonomic mechanical systems of general form is investigated, on the assumption that they have cyclic coordinates and are subject to potential and dissipative forces. A stability theorem, generalizing a theorem proved previously in [1], is established. The problem of the stability of the steady motion of a three-wheeled carriage is considered as an example.

Nonholonomic Mechanics and Control

1R5. Nonholonomic Mechanics and Control. - AM Bloch (Dept of Math, Univ of Michigan, Ann Arbor MI 48109-1109). Springer-Verlag, New York. 2003. 483 pp. ISBN 0-387-95535-6. $69.95.Reviewed by B Brogliato (INRA, 655 Ave De L’Europe, Saint Ismier, 38334, France).This mathematically oriented book is dedicated to the modeling and control of a class of nonlinear mechanical systems, namely mechanical systems subject to nonholonomic (or non integrable) bilateral constraints. It is known that the control of such nonlinear systems requires specific tools, as they may not be stabilizable with continuous feedbacks, as a consequence of Brockett’s necessary conditions. Such problems have received considerable attention in the applied mathematics, mechanics, and systems and control communities, for many years. Therefore, this is a topic of major importance. The audience will mainly consist of graduate students, researchers, either working in this field or with a sufficiently advanced mathematical background (especially differential geometry tools). The book starts with an introductory chapter with many examples treated in detail. The next two chapters are devoted to presenting the mathematical tools needed to study such systems. This material will certainly be hard to follow for those who have no acquaintance in differential geometry and its applications to mechanics. However, in view of the literature on the subject, understanding the basics from geometry seems to be mandatory for someone who wants to understand the control of such mechanical systems. The fourth chapter is dedicated to nonlinear control theory, still with an emphasis on geometry. Controllability, stability and stabilization are reviewed, and the chapter ends with Hamiltonian and Lagrangian control systems. The fifth and sixth chapters are devoted to nonholonomic systems, their dynamics and their control. Stabilization techniques are explained, like time-varying and nonsmooth controllers. Here one can regret that the bibliography omits C Samson, one of the first contributors to the field and who introduced time-varying controllers for the control of chained systems in the paper Velocity and torque feedback control of a nonholonomic cart, in Advanced Robot Control, Proceedings of the International Workshop on Nonlinear and Adaptive Control: Issues in Robotics, Grenoble, France, November 21-23, 1990, Springer Verlag LNCIS 162. Samson’s paper Control of chained systems: Application to path following and time-varying point stabilization, IEEE Trans on Automatic Control, Volume 40, pp 64-77, 1995, would have also been welcome in the bibliography. Chapter 7 deals with optimal control and variational principles. The eighth chapter concerns stability analysis with energy arguments (generalization of the Lejeune-Dirichlet theorem on stability of fixed points). In a logical way the last chapter is devoted to energy-based control and stabilization. There are many other books that cover the topic of nonholonomic mechanical systems control in one or two chapters (for instance Sastry’s book in the same Springer’s series, or the book Theory of Robot Control, Springer CCE Series, 1996). But Nonholonomic Mechanics and Control is entirely dedicated to this topic and covers the aspects of modeling, analysis and control. It clearly belongs to the realm of geometry-oriented works in mechanics or control. The book is well organized, contains many examples and exercises, and can be recommended to all researchers working in the field (applied mathematics, mechanics or control).

Noether symmetry and Hojman conserved quantity for nonholonomic mechanical systems

A nonNoether conserved quantity, i. e. Hojman conserved quantity constructed using the Noether symmetry for the nonholonomic mechanical system is presented. Under special infinitesimal transformations in which the time is not variable, the special Noether symmetry and Noether conserved quantity are given, and the condition under which the special Noether symmetry is a special Lie symmetry is obtained. From the special Noether symmetry, the Hojman conserved quantity of the corresponding holonomic system, the weakly Hojman conserved quantity and the strongly Hojman conserved quantity of the nonholonomic systems are obtained. An example is given to illustrate the application of the result.

Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant manifold. The embedding system possesses a completely natural structure in the context of symplectic geometry, and using it in order to understand properties of the subsystem has compelling advantages. We discuss generic geometric and topological properties of the critical sets of both embedding and physical system, using Conley–Zehnder theory, and by relating the Morse–Witten complexes of the ‘free’ and constrained system to one another. Furthermore, we give a qualitative discussion of the stability of motion in the vicinity of the critical set. We point out key relations to sub-Riemannian geometry, and a potential computational application.

Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant manifold. The embedding system possesses a completely natural structure in the context of symplectic geometry, and using it in order to understand properties of the subsystem has compelling advantages. We discuss generic geometric and topological properties of the critical sets of both embedding and physical system, using Conley–Zehnder theory, and by relating the Morse–Witten complexes of the ‘free’ and constrained system to one another. Furthermore, we give a qualitative discussion of the stability of motion in the vicinity of the critical set. We point out key relations to sub-Riemannian geometry, and a potential computational application.