Formulation and solution for inverse problem of nonholonomic dynamics

This paper presents a formulation and solution for the inverse problem ofnonholonomic dynamics:to find the form of nonholonomic constraints when someintegrals are given and to find the generalized reactive forces of constraint acting onthe system when the expression of the kinetic energy is given.An example is given toillustrate the application of the result.     

Equations of motion for nonholonomic mechanical systems with unilateral constraints

IntroductionThemotionofamechanicalsystemwithunilateralconstraintsismoregeneralthanwithbilateralconstraints1andyetitsinvestigationismoredifficult[1,2].Thesystemswithunilateralconstraintsinvolvetheholonomicmechanicalsystemswithunilateralholonomicconstraints[3～6]andthenonholonondcmeChanicalsystemswithunilateralholonondcconstraintS[7]andthenonholonondcmechanicalsystemswithunilateralnonholonondcconstraintS[2]andsoon.ThispaperstUdiesatypeofmoregeneralsystemswithunilateralconstfaints:nonholonondcs…     

Noether's theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints in event space

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非完整动力学逆问题的基本解法

本文绘出非完整动力学逆问题的两种基本提法和解法,包括运动方程的构成以及运动方程的修改     

On Szebehely's problem extended to holonomic systems With a given integral of motion

Summary We consider a family of curves in the n-dimensional configuration space S_n of a holonomic system with n degrees of freedom. We obtain first-order partial differential equations for the potential function U of forces under which any trajectory belonging to the given family of curves can be described by the representative point of the system. We write the potential function U supposing that, in addition to the energy integral, a first integral of motion linear in the lagrangian velocities is assigned. Next we obtain the compatibility conditions between the energy constant E, the parameter which appears in the first integral, and the n–1 geometric constants c₁, c₂,, c_n–1 which characterize the family of trajectories. Finally we discuss two simple examples.

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Sommario Assegnata una famiglia di curve nello spazio S_n delle configurazioni di un sistema olonomo conservativo ad n gradi di libertà, si determinano le equazioni differenziali alle quali deve soddisfare il potenziale delle forze applicate affinchè il punto rappresentativo del sistema possa descrivere una qualunque delle traiettorie appartenenti alla data famiglia. Si scrive poil' espressione del potenziale supponendo che sia assegnato, oltre l'integrale dell'energia, un integrale primo del moto, lineare nelle velocità lagrangiane. Infine si ricavano le condizioni di compatibilità fra la costante dell'energia E, il parametro che interviene nell'integrale primo e le n-1 costanti geometriche c₁, c₂, c₂,, c_n–1 che caratterizzano la famiglia di curve. Si conclude con due semplici esempi che illustrano quanto esposto.

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Work performed under the auspices of G.N.F.M.-C.N.R., Italy.     

ON A GENERALIZATION OF BERTRAND’S THEOREM

Bertrand’s theorem for the determination of the applied forces to a holonomicsystem from one of its first integrals,is extended to nonholonomic systems.Someinteresting applications of this new result are also given.     

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

ＦＩＲＳＴＩＮＴＥＧＲＡＬＳＡＮＤＩＮＴＥＧＲＡＬＩＮＶＡＲＩＡＮＴＳＦＯＲＶＡＲＩＡＢＬＥＭＡＳＳＮＯＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭＩＮＮＯＮＩＮＥＲＴＩＡＬＲＥＦＥＲＥＮＣＥＦＲＡＭＥＳＬｕｏＳｈａｏ－ｋａｉ（罗绍凯）（Ｓｈａｎｇｑｉｕ,Ｎｏｒ...     

Robust adaptive control of nonholonomic systems with uncertainties

Robust adaptive control of nonholonomic systems in chained form with linearly parameterized and strongly nonlinear disturbance and drift terms is dicussed. The novelty of the proposed method is a combined use of the state-scaling and the back-stepping procedure.     

Integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frames

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

NOETHER’S THEORY FOR NONHOLONOMIC DYNAMICAL SYSTEMS RELATIVE TO NON-INERTIAL REFERENCE FRAME

The new variational principle of Gauss’s form of nonlinear nonholonomicnonpotential system relative to non-inertial reference frame is established byconstructing generalized inertial potentials.Noether’s theorem and Noether’s inversetheorem of the system above is presented and proved.Finally,one example is given toillustrate the application.     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…     

ADAPTIVE REGULATION OF HIGH ORDER NONHOLONOMIC SYSTEMS

The problem of adaptive regulation of a class of high-order parametric non holonomic systems in chained-form was discussed. Using adding a power integrator technique and state scaling with discontinuous projection technique, a discontinuous adaptive dynamic controller was constructed. The controller guarantees the estimated value of unknown parameter is in the prescribed extent.     

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

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

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

KANE’S EQUATIONS FOR PERCUSSION MOTION OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Kane’s equations for the Routh’s form of variable massnonholonomic systems are established.and the Kane’s equations for percussion motionof variable mass holonomic and nonholonomic systems are deduced from them. Secondly,the equivalence to Lagrange’s equations for percussion motion and Kane’sequations is obtained,and the application of the new equation is illustrated by anexample.     

One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

A field method for solving the equations of motion of nonholonomic systems

In this paper, the field method^[1] for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass. Two examples are given to illustrate its application. The project supported by the National Natural Science Foundation of China.     

On the inverse optimization problem for periodic systems

Consideration is given to the problem inverse to the problem of designing the optimal (in the sense of minimization of a quadratic functional) controller for a linear periodic system. Problem statement: given matrices describing the dynamics of the system and a control matrix, determine the weighting matrices of the quadratic functional. To solve this problem, an algorithm based on linear matrix inequalities is proposed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 100–106, December 2005.     

ON THE STABILITY OF NONHOLONOMIC MECHANICAL SYSTEMS WITH RESPECT TO PARTIAL VARIABLES

ＯＮＴＨＥＳＴＡＢＩＬＩＴＹＯＦＮＯＮＨＯＬＯＮＯＭＩＣＭＥＣＨＡＮＩＣＡＬＳＹＳＴＥＭＳＷＩＴＨＲＥＳＰＥＣＴＴＯＰＡＲＴＩＡＬＶＡＲＩＡＢＬＥＳＺｈｕＨａｉ－ｐｉｎｇ（朱海平）ＭｅｉＦｅｎｇ－ｘｉａｎｇ（梅凤翔）（ＢｅｉｊｉｎｇＵｎｉｖｅｒｓｉｔ...     

一类多体系统非完整运动最优规划的拟牛顿算法

研究带有非完整约束的一类多体系统运动规划问题。多体系统中的非完整约束通常是由不可积的速度约束或不可积的守恒律引起。在系统动量和动量矩守恒情况下,动力学方程降阶为非完整形式约束方程,系统的控制问题可转化为无漂移系统的非完整运动规划问题。文中首先导出具有多体开链系统的非完整运动模型。利用最优控制理论和最优化技术,采用输入参数化的方法将连续的最优控制问题转化为离散的最优控制问题,提出一种非完整多体系统运动规划的拟牛顿算法。最后将该方法用于自由漂浮的空间三连杆机构,仿真结果验证了该方法的有效性。