变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量

Herglotz变分原理提供了非保守耗散问题的变分描述，同时变质量力学在自然界和工程领域有大量的应用，因此将Herglotz变分原理应用于变质量力学系统的Lagrange方程与守恒律研究，为研究变质量力学提供了一个新的途径.文中建立了变质量力学系统的Herglotz型广义变分原理，导出了变质量系统的Herglotz型Lagrange方程.定义了变质量力学系统的Herglotz型Noether对称性，建立并证明了Herglotz型Noether定理及其逆定理.文末给出两个变质量非保守系统的具体例子以说明结果的应用.     

事件空间离散完整系统的Noether理论

研究差分离散变分原理和事件空间中离散完整系统的Noether理论. 运用差分离散变分方法,通过群的无限小变换,得到了事件空间中离散完整系统的差分离散变分原理,并建立了离散的运动方程. 得到了系统的Noether对称性的判据方程和Noether守恒量的形式以及其存在的条件. 举例说明结果的应用.     

具有完整约束的变质量系统的机械能守恒律

研究具有双面理想完整约束的变质量力学系统的机械能守恒律.首先,利用系统的运动微分方程,获得能量变化方程,并给出机械能守恒律存在的充分必要条件.然后,提出具有守恒律的27种情形.最后,举例说明.     

有多余坐标的可控完整力学系统的自由运动与初始运动

对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统.本文研究有多余坐标的可控力学系统的自由运动与初始运动.首先,需由d′Alembert Lagrange原理并利用Lagrange乘子法建立有多余坐标完整系统的运动微分方程;其次,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件.第三,如果给定运动的初始条件和控制参数,就可以研究系统的初始运动.文末,举例并说明方法和结果的应用.     

基于球摆模型的离散变分积分子算法研究

在动力学系统长时间的仿真计算中,力学系统固有的结构将影响到计算精度及稳定性.离散变分积分子能够保持力学系统的能量,动量及辛结构的守恒.结合离散变分原理,通过对系统的拉格朗日函数进行离散化以及求变分和积分的过程,可以得到力学系统的离散变分积分子算法.该算法是一种递归算法,给定初始条件便可得到系统的动力学参数的时间历程.使用该原理可以构造具有完整约束的拉格朗日系统的辛-动量积分子方法.与连续算法相比,离散变分积分子算法能够直接在离散拉格朗日函数的基础上得到姿态与角速度的递推公式,而不需要复杂的迭代计算.本文研究是基于第一类拉格朗日函数的离散变分积分子算法.球摆模型是一个具有完整约束的拉格朗日系统.仿真结果表明,系统的能量值在长时间的仿真中得到保持,且计算的精度与步长的数量级呈现二次方的关系,系统角速度和姿态的仿真结果都符合球摆的运动规律.     

关于分析力学的基础与展望

本文从分析约束力学系统的“欠定”问题开始,介绍分析力学的基本变分原理和三类运动微分方程,并分析了分析力学具有普适性之缘由.对非完整约束力学系统,着重分析其动力学建模问题、几何结构和重点发展方向,同时又简要介绍了Birkhoff系统所具有的一般辛结构特征和研究意义,以及需要重点解决的问题.文中对力学系统的Noether对称性和运动微分方程的对称性作了较为详细的论述,并列举了相应实例说明两种对称性与守恒量之间的关系.在几何力学部分,重点介绍了分析力学的辛几何结构和对称性约化理论,包括辛流形的Darboux Moser Weinstein局部正则结构、整体拓扑结构及其对量子力学的影响、Lie群与Lie代数的伴随表示和余伴随表示、动量映射、Cartan辛约化、Marsden Weinstein约化等.文中最后论述了完整与非完整力学系统可积性问题的研究方法和成果,指出了非完整力学系统现有可积性方法的局限性.     

位形空间中约束力学系统的Lagrange对称性与守恒量

研究位形空间中约束力学系统的Lagrange对称性,给出位形空间中约束力学系统的统一动力学方程,给出位形空间中约束力学系统统一方程的Lagrange对称性的判据,得到位形空间中约束力学系统统一方程的Lagrange对称性导致的守恒量及其存在的条件,并举例说明结果的应用.     

非完整力学系统相对运动的稳定性

研究了非完整力学系统相对运动的稳定性.首先,建立了系统的受扰运动微分方程,进而推导了系统的能量变化方程;其次,基于能量变化方程,给出了非完整力学系统相对运动的稳定性的一个判据;最后,举例说明结果的应用.     

本质线性非完整系统的Hamilton原理

研究了本质线性非完整系统的Hamilton原理,分别应用与不应用Appell-Chetaev条件证明了本质线性非完整系统Hamilton变分泛函取驻值的充分必要条件.结果表明,在本质线性非完整系统中,Hamilton作用量是稳定的作用量,与完整系统的Hamilton原理具有相同的形式与本质;而且由Hamilton原理得到的运动方程不会导致任何力学与数学上的矛盾.最后给出了Hamilton原理向本质非线性非完整系统推广时产生数学与力学上不合理的根本原因.     

关于Gauss原理

Gauss原理是分析力学中的一个微分变分原理,它在理论上简单,应用上有优势,而且适用于双面理想完整系统和非完整系统.本文对这个原理的形成和发展给出一些史料,并提出一些看法.     

本质线性非完整系统的Hamilton的原理

研究了本质线性非完整系统的Hamilton原理，分别应用与不应用Appell—Chetaev条件证明了本质线性非完整系统Hamilton变分泛函取驻值的充分必要条件．结果表明，在本质线性非完整系统中，Hamilton作用量是稳定的作用量，与完整系统的Hamilton原理具有相同的形式与本质；而且由Hamilton原理得到的运动方程不会导致任何力学与数学上的矛盾．最后给出了Hamilton原理向本质非线性非完整系统推广时产生数学与力学上不合理的根本原因。     

Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top

This paper studies local configuration controllability of multibody systems with nonholonomic constraints. As a nontrivial example of the theory, we consider the dynamics and control of a multibody spherical robot. Internal rotors and sliders are used as the mechanisms for control. Our model is based on equations developed by the second author for certain mechanical systems with nonholonomic constraints, e.g. Chaplygin's sphere and Chaplygin's top in particular, and the multibody framework for unconstrained mechanical systems developed by the first and third authors. Recent methods for determining controllability and path planning for multibody systems with symmetry are extended to treat a class of mechanical systems with nonholonomic constraints. Specific results on the controllability and path planning of the spherical robot model are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Control design based on dead-zone and leakage adaptive laws for artificial swarm mechanical systems

We consider the control design of artificial swarm systems with emphasis on four characteristics. First, the agent is made of mechanical components. As a result, the motion of each agent is subject to physical laws that govern mechanical systems. Second, both nonlinearity and uncertainty of the mechanical system are taken into consideration. Third, the ideal agent kinematic performance is treated as a desired d’Alembert constraint. This in turn suggests a creative way of embedding the constraint into the control design. Fourth, two types of adaptive robust control schemes are designed. They both contain leakage and dead-zone. However, one design suggests a trade-off between the amount of leakage and the size of dead-zone, in exchange for a simplified dead-zone structure.     

Invariant solutions and conservation laws to nonconservative FP equation

We generate conservation laws for the one dimensional nonconservative Fokker–Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker–Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries.     

开链机械臂的反馈Hamel变分积分子

在几何力学框架下,首先,推导了开链机械臂的反馈Hamel变分积分子,其中用反馈控制算法实现了Hamel积分子的寻根;其次,将上述算法用于计算反应规划的正向动力学,在此基础上将回转力引入传统人工势场法来设计斥力,可以克服传统人工势场法中易陷入势场局部极小的缺点,实现机械臂整臂的实时避障轨迹规划;最后,通过两个数值仿真验证了上述所提算法的有效性.     

Sequential circle criterion approach for robustly stabilizing individual generators with SVC

Absolute stability with the spatially defined linear time‐invariant (LTI) state‐space modelings is scrutinized by means of what we call the sequential Lyapunov approach, which possesses independent significance in stabilization when gain‐scheduling control laws are adopted. Then, this theoretical result is exploiting for stabilization of individual generators via SVC actions. More precisely, by remodeling the perturbed swing equations of synchronous generators in multimachine networks through spatially defined LTI state‐space expressions subjected to uncertainties and power disturbance, which are viewed as sector nonlinearities, we introduce frequency responses for coping with nonlinear power swing dynamics of individual generators. By sequentially relating the frequency responses to the circle criterion (substantially, the KYP theorem or the positive real lemma) claimed for LTI systems subject to sector disturbances, output feedback control laws for static VAR compensators are worked out to stabilize individual generators. The frequency‐domain approach is also useful in steady‐state specification besides stabilization in individual generators. Examples show efficacy of the suggested stabilization and steady‐state specification technique. Copyright © 2014 John Wiley & Sons, Ltd.     

The dynamics of a forced sphere-plate mechanical system

We study the dynamics and explore the controllability of a family of sphere-plate mechanical systems. These are nonholonomic systems with a five-dimensional (5-D) configuration space and three independent velocities. They consist of a sphere rolling in contact with two horizontal plates. Kinematic models of sphere-plate systems have played an important role in the control systems literature addressing the kinematics of rolling bodies, as well as in discussions of nonholonomic systems. However, kinematic analysis falls short of allowing one to understand the dynamic behavior of such systems. We formulate and study a dynamic model for a class of sphere-plate systems in order to answer the question: -is it possible to impart a net angular momentum to a sphere which rolls without slipping between two plates, given that the position of the top plate is subject to exogenous forces?”     

一类非完整系统的全局镇定

当非完整系统只能局部转换为链式形式时, 由于存在变换奇异点集合, 针对链式系统所设计的全局反馈控制律只能局部镇定原非完整系统, 而且当期望状态接近奇异点时, 闭环系统的吸引区很小. 本文针对一类可局部转换为链式系统的非完整系统, 首先利用吸引区是状态空间中的一个不变集且与变换奇异点集不相交的条件导出了一个吸引区的不变子集, 然后给出了将系统状态从任意点驱动到吸引区不变子集内的开环控制算法, 最后结合开环控制和闭环控制得到一种混合控制算法. 该混合控制算法可以保证任意不在变换奇异点集合内的期望状态是全局渐近稳定的. 对平面两转动关节空间机器人的仿真结果证实了算法的有效性.     

基于积分因子方法研究Chaplygin非完整系统的守恒律

提出并研究了构建Chaplygin非完整系统守恒律的积分因子方法.基于正则形式的Chaplygin方程,定义了积分因子,给出了系统存在守恒量的必要条件,建立了Chaplygin非完整系统的守恒定理及其逆定理.研究表明:对应于必要条件的每一组非奇异函数解,系统存在一个守恒量;反之,对于一个已知守恒量,可找到相应的积分因子,且解是不唯一的.文末以匀质圆球在粗糙水平面上纯滚动为例,讨论了该方法的应用.