几何代数的高阶逻辑形式化

几何代数是一种用于描述和计算几何问题的代数语言.由于它统一的表达分析和不依赖于坐标的几何计算等优点现已成为数学分析、理论物理、几何学、工程应用等领域重要的理论基础和计算工具, 然而利用几何代数进行计算和建模分析的传统方法,如数值计算方法和符号方法等都存在计算不精确或者不完备等问题.高阶逻辑定理证明是验证系统正确的一种严密的形式化方法.本文在高阶逻辑证明工具HOL-Light中建立了几何代数系统的形式化模型,主要包括片积、多重矢量、外积、内积、几何积、几何逆、对偶、基矢量运算和变换算子等的形式化定义和相关性质定理的证明.最后为了说明几何代数形式化的有效性和实用性,本文在共形几何代数空间中对刚体运动问题提供了一种新的简单有效的形式化建模与验证方法.     

有向网络下线性多个体系统的整体行为—–矩阵方法

研究了有向通讯网络条件下一阶和二阶线性多个体协同动力学系统整体行为的矩阵代数性质.利用矩阵分析的方法将系统的系数矩阵变换为Frobenius标准型,由此将系统分解为独立基本子系统和非独立基本子系统的组成结构.通过研究行和为零的对角占优矩阵的性质,得出了对线性多个体协同动力学系统整体行为起决定作用的系数矩阵的性质,从而将这一问题转换为普通线性代数问题.     

摄动开普勒问题形式化建模与验证

摄动开普勒问题广泛应用于卫星轨道摄动分析,然而卫星轨道摄动分析数学模型的错误将导致灾难性后果.传统的建模与分析方法涉及到矢量代数、旋量代数、复数、四元数等多种不同的代数系统,在各个代数系统相互转换过程中极易引入错误.几何代数方法将多种代数系统统一到相同代数结构中,弥补了传统分析方法的不足.但是基于几何代数的摄动开普勒问题数学模型的正确性并没有通过严格的形式化验证.本文采用高阶逻辑来描述该问题的属性和规范,以公认的逻辑公理和推理规则为基础构建其形式化模型并进行验证,从而最大程度确保数学模型的正确性和分析方法的可靠性.     

基于Lie理论对两轮不稳定小车的变结构控制

为了有效控制两轮不稳定小车,减少在控制过程中产生的抖振,首先利用一种非线性系统微分几何方法-Lie理论对两轮不稳定小车系统进行坐标和输入量的变换,实现了系统的局部反馈线性化,进而通过近似得到系统的线性模型.然后针对常规滑模变结构控制抖振较强的问题,设计了一种新的滑模控制方法,用动态滑模变结构控制.方法通过设计新的切换函数,使切换函数与系统控制输入的一阶或高阶导数有关,可将不连续项转移到控制的一阶或高阶导数中去,得到在时间上本质连续的动态滑模控制律,有效地降低了抖振.仿真和实验结果表明,采用的方法以及设计的控制器对不稳定小车的控制是有效的.     

非匹配不确定高阶非线性系统递阶Terminal滑模控制

对于高阶非线性系统,首先采用改进的高阶滑模微分器作为间接干扰观测器,获得前n-1个子系统中的非匹配复合干扰的估计值,证明了估计误差可任意小. 为避免代数环,设计了三种方案获得最后一个子系统中非匹配复合干扰的估计值,并证明了估计误差有界. 在此基础上设计递阶Terminal滑模控制器,证明了控制器参数非奇异及结构非奇异,并给出所需条件. 最后,证明了系统稳定,跟踪误差可任意小. 近空间飞行器姿态控制仿真验证了本文结论.     

分数阶动力学的分析力学方法及其应用

综述我们在分数阶动力学分析力学方法的研究进展,包括：分数阶动力学系统的分析力学表示,构造分数阶动力学模型的分析力学方法,构造分数阶动力学模型团簇的分析力学方法,三类分数阶Lie群无限小变换方法,分数阶动力学系统的对称性、对称性摄动和共形不变性的分析力学方法,分数阶动力学系统的代数结构与Poisson积分的分析力学方法,构造分数阶动力学系统积分不变量的分析力学方法,分数阶动力学系统梯度表示的分析力学方法,分数阶动力学系统稳定性的分析力学方法,分数阶微分方程的分析力学方法等,介绍了对于物理学、力学、生物学、非线性科学等领域的10多种分数阶动力学模型的应用,并指出了若干进一步研究的问题.     

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.     

一阶系统自伴随条件和Lagrange函数的构造

重新研究一阶系统变分法逆问题,修正了某些文献关于一阶微分方程组自伴随条件的失误,导出了把一阶系统化为自伴随形式的变换矩阵所满足的方程,列出了构造自伴随一阶系统Lagrange函数的两种方法.举例说明所得结果的应用.     

不同阶外激励下压电纤维复合材料悬臂板的内共振特性分析

本文研究了不同阶横向激励作用下压电纤维复合材料(MFC)悬臂板的非线性动力学特性.基于Reddy一阶剪切板理论,引入von Kármán几何非线性理论,利用Hamilton原理建立了结构的非线性动力学控制方程.讨论在不同结构尺寸下的前三阶固有频率,利用Galerkin方法将系统离散为三自由度的非线性常微分方程.考虑主参数共振-1:3:5内共振,分析不同阶外激励作用下压电纤维复合悬臂板的非线性振动响应.数值模拟结果表明,复合材料板几何尺寸增大时结构的固有频率降低.此外,不同阶的横向激励幅值对结构的非线性振动影响很大,为实际工程应用提供理论支持.     

非线性系统的反演自适应动态滑模控制

为削弱非线性系统中由滑模控制引起的抖振现象,提出自适应动态滑模控制方案.动态滑模方法是通过设计新的切换函数或将常规滑模变结构控制中的切换函数s通过微分环节构成新的切换函数,该切换函数与系统控制输入的一阶或高阶导数有关,可将不连续项转移到控制的一阶或高阶导数中去,得到在时间上本质连续的动态滑模控制律.该方案利用自适应技术和反演法,设计动态滑模控制器,有效地削弱了系统的抖振.最后仿真结果证明了该方案是正确的.     

Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations

This paper is concerned with the stabilization of nonholonomic systems in port-controlled Hamiltonian formulae based on time-varying generalized canonical transformations. A special class of time-varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity. Utilizing these transformations, time-varying asymptotically stabilizing controllers for the nonholonomic Hamiltonian systems are derived. Since the proposed method is a natural generalization of passivity based control for conventional holonomic systems, it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method.     

Advanced Programmed Motion Tracking Control of Nonholonomic Mechanical Systems

Trajectory tracking control of nonholonomic systems has been extended to tracking a desired motion. The desired motion is specified by equations of constraints, referred to as programmed, which may be differential equations of high order and may be nonholonomic. The strategy enables motion tracking control under the assumption that the system dynamics are accurately known. It is referred to as a model reference tracking control strategy for programmed motion. In this paper, adaptive and repetitive extensions of the strategy are proposed. Two selected advanced tracking control algorithms, i.e., the desired compensation adaptation law and the repetitive control law, which were originally dedicated to holonomic systems, are adapted to motion tracking control of nonholonomic systems. Simulation studies that illustrate programmed motion tracking control of systems with unknown parameters and the performance of repetitive motions are provided. A new performance measure to evaluate a programmed motion tracking performance is introduced.      

Nonholonomic Mobile Manipulators: Kinematics, Velocities and Redundancies

We consider nonholonomic mobile manipulators built from an n _a joint robotic arm and a nonholonomic mobile platform with two independently driven wheels. Actually, there is no efficient kinematic formalism for these systems which are generally characterized by their high number of actuators. So, kinematic modelling is presented with particular emphasis on redundancy. Whereas kinematic redundancy is well known in the holonomic case, it is pointed out that it is necessary to define velocity redundancy in the case of nonholonomic systems. Reduced velocity kinematics based on quasi-velocities are shown to provide an efficient formalism. Two examples of mobile manipulators are presented. Finally, reduced velocity kinematics and velocity redundancy are shown to be adequate tools in order to realize operational task while optimizing criteria such as manipulability.     

混合非完整系统问题

研究一类混合非完整系统的运动．它可分为3个阶段：第1阶段为完整系统的连续运动,第2阶段为冲击运动,第3阶段为非完整系统的连续运动．后一阶段的初始条件由前一阶段的运动终了条件确定．举例说明结果的应用．     

Coordinating mobility and manipulation using nonholonomic mobile manipulators

This paper deals with coordinated tasks for mobile nonholonomic manipulators. The systems are composed of a nonholonomic mobile platform and a simple kinematic chain holonomic arm. First, concepts such as redundancy and singular configurations are defined. Then, reduced differential models are introduced. It is then possible to work with a set of independent task coordinates. Applications follow for a planar system in the case of point-to-point and continuous path tasks.     

经典非完整Appell Hamel约束的性质

研究了非完整约束Appell-Hamel例.证明了经典Appell-Hamel例对于非完整系统的Hamilton作用量是稳定值.研究了该约束对于非完整力学Rosen-Edelstein模型的解,证明了对于三个非完整力学模型Ap-pell-Hamel例具有相同解.利用非完整力学系统可归结为有条件的完整系统的理论,得出了经典Appell-Hamel例具有第二类Lagrange方程的形式.     

Robust motion tracking control of partially nonholonomic mechanical systems

This paper proposes a stable motion tracking control law for mechanical systems subject to both nonholonomic and holonomic constraints. The control law is developed at the dynamic level and can deal with model uncertainties. The proposed control law ensures the desired trajectory tracking of the configuration state of the closed-loop system. A global asymptotic stability result is obtained in the Lyapunov sense. A detailed example is presented to illustrate the proposed method.     

Robust tracking designs for both holonomic and nonholonomicconstrained mechanical systems: adaptive fuzzy approach

Adaptive fuzzy-based tracking control designs are proposed in the paper for both holonomic mechanical systems as well as a large class of nonholonomic mechanical systems with plant uncertainties and external disturbances. A unified and systematic procedure is employed to derive the controllers for both holonomic and nonholonomic mechanical control systems, respectively. First, a fuzzy logic system is introduced to learn the behavior of unknown (or uncertain) mechanical dynamics by using an adaptive algorithm. Next, the effect of approximation error on the tracking error must be efficiently eliminated by employing an additional robustifying algorithm. Consequently, hybrid adaptive-robust controllers can be constructed such that the resulting closed-loop mechanical systems guarantee a satisfactorily transient and asymptotic performance. Furthermore, a partitioned procedure with respect to the above developed adaptive fuzzy logic approximators is introduced such that the number of fuzzy IF-THEN rules is significantly reduced and the developed control schemes can be easily implemented from the viewpoint of practical applications. Finally, simulation examples are presented to illustrate the tracking performance of a two-link constrained robot manipulator and a vertical wheel rolling on a plane surface by the proposed adaptive fuzzy-based control algorithms     

事件空间中非完整力学系统的Herglotz-d′Alembert原理与守恒律

研究事件空间中非完整力学系统的Herglotz型守恒律.给出事件空间中Herglotz型广义变分原理,引入非完整约束并采用交换关系的H9lder定义,导出事件空间中非完整力学系统的新型微分变分原理—Herglotz-d′Alembert原理.引进事件空间中的空间生成元和参数生成元,建立Herglotz-d′Alembert原理不变性条件的变换.基于该原理构建了事件空间中非完整非保守力学系统的Herglotz型守恒定理及其逆定理.作为特例,给出了位形空间的Herglotz型守恒量和事件空间中完整力学系统的Herglotz型守恒量.文末还给出了一个算例.     

A New High‐Order Adaptive Robust Control for Constraint Following of Mechanical Systems

A mechanical system is to follow a class of prescribed holonomic or nonholonomic constraints. The system contains time‐varying bounded uncertainty. However, the bound is unknown. The objective is to design a control which renders constraint following. A new high‐order adaptive robust control is proposed. The control guarantees uniform boundedness and uniform ultimate boundedness even in the presence of the uncertainty. Significant advantages of this new control are demonstrated both analytically and numerically. It is shown that the system performance, including the finite entering time, constraint‐following error, and control magnitude, can be improved by tuning the control order.