非完整约束系统几何动力学研究进展:Lagrange理论及其它

近10年来, 非完整力学的发展主要集中在两个相互关联的方向上, 一个是非完整运动规划, 另一个则是非完整约束系统的几何动力学, 这两个研究方向都充分地利用了现代几何学, 如纤维丛理论、辛流形和Poisson流形结构等等.本文主要综述非完整约束系统几何动力学的外附型和内禀型Lagrange理论, 包括非定常力学系统所需要的射丛几何学的基本概念、射丛按约束的直和分解、约束流形上的水平分布、D'Alembert-Lagrange方程与Chaplygin方程的整体描述、以及Riemann-Cartan流形上的非完整力学, 文中对Chetaev条件和d-δ交换关系的几何意义作了深入讨论.除此之外, 简要评述非完整力学的Hamilton理论与赝Poisson结构、Noether对称性和Lie对称性、动量映射与对称约化、Vakonomic动力学等几个非常重要专题的研究进展.      

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied. The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed. The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system. An example is given to illustrate the application of the result.     

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。     

动态规划求解空间双臂机器人非完整运动最优控制问题

通过自适应动态规划研究自由漂浮空间双臂机器人运动的最优控制问题．针对空间双臂机器人的非完整性运动,采用自适应动态规划(Adaptive Dynamic Programming, ADP)方法求解其最优控制问题．根据多体动力学理论,推导了载体位置、姿态均无控制条件下,双臂空间机器人满足的系统动量守恒关系的非完整约束方程,并将其转化为控制系统的状态方程,从而将双臂空间机器人的非完整运动规划问题转化为对非线性系统的控制问题．文中根据自适应动态规划网络结构,利用神经网络来近似性能指标函数,进而用龙格库塔法求解状态变量．并给出了适合该类问题的一种效用函数具体表达式,保证了空间双臂机器人到达期望位置后不再继续运动．实现了对空间双臂机器人非完整运动规划的最优控制．数值仿真实验验证了ADP对求解空间双臂机器人非完整运动规划最优控制问题的有效性．     

whittaker’s Reduction Method for Poincare’s Dynamical Equations

ＷＨＩＴＴＡＫＥＲ＇ＳＲＥＤＵＣＴＩＯＮＭＥＴＨＯＤＦＯＲＰＯＩＮＣＡＲＥ＇ｓＤＹＮＡＭＩＣＡＬＥＱＵＡＴＩＯＮＳＱ．Ｋ．Ｇｈｏｒｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅｓ，ＫｉｎｇＦａｈｄＵｎｉｖｅｒｓｉｔｙｏｆＰｅｔｒｏ...     

ON THE NONHOLONOMIC CONSTRAINTS AND MOTION CONTROL OF WHEELED MOBILE STRUCTURES1)

当质点沿光滑曲线运动时,必须满足横向速度为零的条件.同样地,不同轮式移动结构在平面上做光滑曲线运动时都需要满足该非完整约束条件.本文结合轮子转速和它们运动速度的完整约束关系,理清各轮式移动结构的完整和非完整约束,然后利用 Euler-Lagrange 方程方便地推导出相应的动力学方程.另外,通过该非完整约束,将目标轨迹曲线转化为速度目标的形式,然后引入目标轨迹曲线的相对曲率设计合适的动态跟踪目标.进一步,通过采用该动态跟踪目标可以将轮式移动结构的运动规律和动力学方程有机结合起来,并将原运动任务简化为一般的 轨迹 控制问题.基于该动态跟踪目标可以为轮式移动结构设计合适的鲁棒跟踪控制器,通过跟踪目标轨迹曲线的相对曲率来实现对目标曲线的精确跟踪.最后,理论分析和仿真结果显示,采用动态目标跟踪方法能够从根本上解决初始速度误差过大和位置误差不断被累积的问题,即使前向速度误差系统不稳定的,实际运动轨迹也几乎能和目标轨迹曲线重合.     

Whittaker's reduction method for poincaré's dynamical equations

Whittaker's reduction method invokes the energy integral to reduce the order of Lagrange's equations of motion of a holonomic dynamical system. This paper treats the corresponding results for a nonholonomic conservative system described by Poincare's equations which are constructed from the standpoint of the theory of Lie groups.     

Form invariance and conserved quantity for weakly nonholonomic system

The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.     

THE LIE SYMMETRIES AND CONSERVED QUANTITIES OF VARIABLE-MASS NONHOLONOMIC SYSTEM OF NON-CHETAEV''''S TYPE IN PHASE SPACE

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｓｙｍｍｅｔｒｙａｎｄｃｏｎｓｅｒｖｅｄｑｕａｎｔｉｔｙｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍｉｓａｎｉｍｐｏｒｔａｎｔｔｏｐｉｃｉｎｍａｔｈｅｍａｔｉｃｓ,ｍｅｃｈａｎｉｃｓａｎｄｐｈｙｓｉｃｓ .ＴｈｅｍｏｄｅｒｎｔｈｅｏｒｉｅｓｏｆｓｙｍｍｅｔｒｙａｎｄｃｏｎｓｅｒｖｅｄｑｕａｎｔｉｔｙｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍｉｎｃｌｕｄｅＮｏｅｔｈｅｒｓｙｍｍｅｔｒｙｔｈｅｏｒｙａｎｄＬｉｅｓｙｍｍｅｔｒｙｔｈｅｏｒｙ .Ｉｎ 1979Ｍ .Ｌｕｔｚｋｙａｎｄｏ…     

An extension of Levi-Civita's theorem to nonholonomic dynamical systems

This paper uses Poincaré formalism to extend the Levi-Civita theorem to cope with nonholonomic systems admitting certain invariant relations whose equations of motion involve constraint multipliers. Sufficient conditions allowing such extension are obtained and, as an application of the theory a generalization of Routh's motion is presented.     

On the invariant theory of nonholonomic systems with constraints of non-Chetaev type

This paper deals with the theory of the differential invariant and integral invariant for a nonholonomic system with constraints of non-Chetaev type. It gives the restricted conditions of virtual displacement in velocity space for nonholonomic constraints of non-Chetaev type and extends the Jourdain principle and the canonical equation for the system. It presents and proves generalized Noether theorem, and gives generalized energy integral and cyclic integral for the system. Finally, the basic integral variants for the system are extended and the integral invariant of Poincaré-Cartan and the universal integral invariant of Poincaré for the system are obtained.     

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.     

Szebehely's problem for nonholonomic systems with a given first integral

I.IntroductionTheinverseproblemofdynamicsisoneoftheimportantsubjectsinmechanics.In1977,Szebehelysetforthaninverseproblemforthedeterminationofthet'orcefunctiontoamaterialpointintheplanefromparametricfamilyoftrajectories,andobtainedalinearfirstorderpartialdifferentialequationfortheforcefunction.Later,Erdil'l,MellsandPirast=l,MellsandBorgherol'l,BoilsandMertnsl4]extendedSzebehely'sproblemtoboththreeandndimensionalholonomicsystem.Recently,theauthorandProfessorMetFengxiangl'1studiedtheSzebehe…     

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 massnonholonomic nonconservative system in noninertial reference frame, proves theequivalence between Holder form and Suslov form, and then obtains differential equationsof motion of variable mass nonholonomic nonconservative system in noninertial referenceframe.     

Lie symmetries and conserved quantities of second-order nonholonomic mechanical system

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly , the inverse problems of the Lie symmetries are studied . Finally , an example is given to illustrate the application of the result.     

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

Ｔｈｅｓｔｕｄｙｏｆｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｄｙｎａｍｉｃａｌｓｙｓｔｅｍｉｓａｓｉｇｎｉｆｉｃａｎｔａｓｐｅｃｔｏｆｍｏｄｅｒｎａｎａｌｙｔｉｃａｌｍｅｃｈａｎｉｃｓ.Ｉｎ1918,ｕｓｉｎｇｔｈｅｉｎｖａｒｉａｎｃｅｏｆＨａｍｉｌｔｏｎａｃｔｉｏｎｕｎｄｅｒｔｈｅｉｎｆｉｎｉｔｅｓｉｍａｌｔｒａｎｓｆｏｒｍａｔｉｏｎｓ,Ｎｏｅｔｈｅｒ[1]ｓｔｕｄｉｅｄｃｏｎｓｅｒｖａｔｉｏｎｌａｗｓｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍａｎｄｐｒｅｓｅｎｔｅｄｔｈｅｆａｍｏｕｓＮｏｅｔｈｅｒ’ｓｔｈｅｏｒ…     

包含伺服约束的非完整系统的对称性摄动与绝热不变量

建立包含伺服约束系统的运动微分方程,基于包含伺服约束系统对称性与不变量理论研究包含伺服约束系统的对称性摄动与绝热不变量 问题,提出了包含伺服约束系统的高阶绝热不变量概念,证明了精确不变量与绝热不变量存在的条件及形式,并举例说明结果的应用。     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

具有单面非线性非完整约束的动力学系统的运动

本文提出并研究了具有单面非线性非完整约束的动力学系统的运动问题,给出了描述系统运动的Ｂｏｌｔｚｍａｎｎ－Ｈａｍｅｌ型方程,并举例说明结果的应用。