A DIFFERENTIAL GEOMETRIC DESCRIPTION FOR TIME-INDEPENDENT CHETAEV'S NONHOLONOMIC MECHANICAL SYSTEM WITH UNILATERAL CONSTRAINTS

By applying the framework of the tangent bundle geometry to the method of Lagrange multi- pliers,a geometric description of Chetaev's nonholonomic systems subjected to unilateral nonholonomic con- straints trod unilateral holonomic constraints respectively in time-independent circumstances is presented.

Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints

Based on the dynamical theory of multi-body systems with nonholonomic constraints and an algorithm for complementarity problems, a numerical method for the multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints is presented. In particular, a wheeled multi-body system is considered. Here, the state transition of stick-slip between wheel and ground is transformed into a nonlinear complementarity problem （NCP）. An iterative algorithm for solving the NCP is then presented using an event-driven method. Dynamical equations of the multi-body system with holonomic and nonholonomic constraints are given using Routh equations and a con- straint stabilization method. Finally, an example is used to test the proposed numerical method. The results show some dynamical behaviors of the wheeled multi-body system and its constraint stabilization effects.

Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints

Lie symmetry and conserved quantity deduced from Lie symmetry of Appell equations in a dynamical system of relative motion with Chetaev-type nonholonomic constraints are studied.The differential equations of motion of the Appell equation for the system,the definition and criterion of Lie symmetry,the condition and the expression of generalized Hojman conserved quantity deduced from Lie symmetry for the system are obtained.The condition and the expression of Hojman conserved quantity deduced from special Lie symmetry for the system under invariable time are further obtained.An example is given to illustrate the application of the results.

Lie symmetry and Hojman conserved quantity of a Nielsen equation in a dynamical system of relative motion with Chetaev-type nonholonomic constraint

The Lie symmetry and Hojman conserved quantity of Nielsen equations in a dynamical system of relative motion with nonholonomic constraint of the Chetaev type are studied. The differential equations of motion of the Nielsen equation for the system, the definition and the criterion of Lie symmetry, and the expression of the Hojman conserved quantity deduced directly from the Lie symmetry for the system are obtained. An example is given to illustrate the application of the results.

Unified symmetry of mechano-electrical systems with nonholonomic constraints

The unified symmetry of mechano-electrical systems with nonholonomic constraints are studied in this paper, the definition and the criterion of unified symmetry of mechano-electrical systems with nonholonomic constraints are derived from the Lagrange-Maxwell equations. The Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity are then deduced from the unified symmetry. An example is given to illustrate the application of the results.

A METHOD FOR SOLVING THE DYNAMICS OF MULTIBODY SYSTEMS WITH RHEONOMIC AND NONHOLONOMIC CONSTRAINTS

The solution of the dynamic problem of multibody systems subject to rheonomic and nonholonomic constraints is achieved by applying singular value decomposition of the constraint matrix and projections of the dynamic equations of the systems along the feasible and unfeasible directions of the constraints. Formula to solve the constraint reaction forces and a method to avoid the violation of the constraints are also given.The solution does not rely on coordinates used to describe the systems and is computational efficitive example is finally presnted.

Intelligent Integrated Control of Underactuated Mechanical Systems with Second-order Nonholonomic Constraints

This paper describes an intelligent integrated control of an acrobot, which is an underactuated mechanical system with second-order nonholonomic constraints. The control combines a model-free fuzzy control, a fuzzy sliding-mode control and a model-based fuzzy control. The model-free fuzzy controller designed for the upswing ensures that the energy of the acrobot increases with each swing. Then the fuzzy sliding-mode controller is employed to control the movement that the acrobot enters the balance area from the swing-up area. The model-based fuzzy controller, which is based on a Takagi-Sugeno fuzzy model, is used to balance the acrobot. The stability of the fuzzy control system for balance control is guaranteed by a common symmetric positive matrix, which satisfies linear matrix inequalities.

Mei symmetry and new conserved quantities of Tzénoff equations for higher-order nonholonomic system

In this paper,the Mei symmetry of Tzénoff equations for the higher-order nonholonomic system and the new conserved quantities derived from that are researched,and the function expression of new conserved quantities and criterion equation which deduces these conserved quantities are presented.This result establishes the theory basis for further researches on conservation laws of Tzénoff equations of the higher-order nonholonomic constraint system.

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

The first integrals and their conditions of existence for variable massnonholonomic system in noninertial relerence frames are obtained,and the canonicalequations and the variation equations of the system are extended. It is proved that using the first integral we can construct the integral invariant of the system.Finally,a series of deductions and an example are given.

Anomaly detection method based on kinematics model and nonholonomic constraint of vehicle

A method used to detect anomaly and estimate the state of vehicle in driving was proposed.The kinematics model of the vehicle was constructed and nonholonomic constraint conditions were added,which refer to that once the vehicle encounters the faults that could not be controlled,the constraint conditions are violated.Estimation equations of the velocity errors of the vehicle were given out to estimate the velocity errors of side and forward.So the stability of the whole vehicle could be judged by the velocity errors of the vehicle.Conclusions were validated through the vehicle experiment.This method is based on GPS/INS integrated navigation system,and can provide foundation for fault detections in unmanned autonomous vehicles.

INTEGRAL THEORY FOR THE DYNAMICS OF NONLINEAR NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

This paper establishes the integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frame. Firstly, based on the Routh equation of the relative motion of nonlinear nonholonomic system gives the first integral of the system. Secondly, by using cyclic integral or energy integral reduces the order of the equation and obtains generalized Routh equation and Whittaker equation respectively. Thirdly, derives canonical equation and variation equation and by using the first integral constructs integral invariant. And then, establishes the basic integral variants and the integral invariant of Poincare-Cartan type. Finally, we give a series of deductions.

CONSERVATION LAWS OF NONHOLONOMIC NONCONSERVATIVE DYNAMICAL SYSTEMS

In this paper conservation laws of nonholonomic nonconservative dynamical systems are studied by using the differential variational principles of Jourdain and the generalized Noether's identi- ties of nonconservative systems subject to first order nonlinear nonholonomic constraints are provided.

THE PRINCIPLES OF LEAST ACTION OF VARIABLE MASS NONHOLONOMIC NONCONSERVATIVE SYSTEM IN NONINERTIAL REFERENCE FRAMES

This paper presents the generalized principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frame, proves the equivalence between Holder form and Suslov form, and then obtains differential equations of motion of variable mass nonholonomic nonconservative system in noninertial reference frame.

INTEGRATION METHOD FOR THE DYNAMICS EQUATION OF RELATIVE MOTION OF VARIABLE MASS NONLINEAR NONHOLONOMIC SYSTEM

In this paper, the integration methods of dynamics equations of relative motion of variable mass nonlinear nonholonomic system are given such as the gradient method, the single-component method and the field method. Firstly, the dynamics equations are written in the canonical form and the field form. Secondly, the gradient method, the single-component method and the field method are used to integrate the dynamics equations of the corresponding constant mass holonomic system in inertial reference frame respectively. With the restriction of nonholonomic constraints to the initial conditions being considered, the solutions of the dynamics equations of variable mass nonlinear nonholonomic system in noninertial reference frame are obtained.

Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces

The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative, conservative, and circulatory forces. The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity. It is assumed that the kinetic energy, the Rayleigh dissipation function, and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions. The results obtained here are partially generalized the results obtained by Kozlov et al. （Kozlov, V. V. The asymptotic motions of systems with dissi- pation. Journal of Applied Mathematics and Mechanics, 58（5）, 787-792 （1994）. Merkin, D. R. Introduction to the Theory of the Stability of Motion （in Russian）, Nauka, Moscow （1987）. Thomson, W. and Tait, P. Treatise on Natural Philosophy, Part I, Cambridge University Press, Cambridge （1879））. The results are illustrated by an example.

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper, Routh 's 'equations for the mechanical systems of the variable mass with nonlinear nonholonomic constraints of arbitrary orders in a noninertial reference system have been deduced not from any variational principles, but from the dynamical equations of Newtonian mechanics. And then again the other forms of equations for nonholonomic systems of variable mass are obtained from Routh's equations.

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.

Vibration-Driven Microrobot Positioning Methodologies for Nonholonomic Constraint Compensation

This paper presents the formulation and practical implementation of positioning methodologies that compensate for the nonholonomic constraints of a mobile microrobot that is driven by two vibrating direct current(DC) micromotors. The open-loop and closed-loop approaches described here add the capability for net sidewise displacements of the microrobotic platform. A displacement is achieved by the execution of a number of repeating steps that depend on the desired displacement, the speed of the micromotors, and the elapsed time. Simulation and experimental results verified the performance of the proposed methodologies.

GENERALIZED NOETHER’S THEOREM OF NONHOLONOMIC NONPOTENTIAL SYSTEM IN NONINERTIAL REFERENCE FRAMES

The new Lagrangian of the relative motion of mechanical system is constructed, the varialional principles oj Jourdain's form of nonlinear nonlwlonomic nonpotential system in noninertial reference frame are established, the generalized Noether's theorem of the system above is presented and proved, and the conserved quantities of system are studied.

THE INTEGRATION METHODS OF VACCO DYNAMICS EQUATION OFNONLINEAR NONHOLONOMIC SYSTEM

This paper presents the integration methods for vacco dynmmies equations of nonlinear nonholononic system,First.vacco dynamies equations are written in the canonical form and the field form.second the gradient methods the single-componentmethods and the field method are used to integrate the dynamics equations of the corresponding holonomic system respectively.And considering the restriction of nonholonomic construint to the initial conditions the solutions of Vacco dynamics cquations of nonlinear nonholonomic system are obtained.