机构学及优化设计中基于混沌分形的理论与方法的研究
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摘要
混沌分形作为非线性科学研究的核心内容，近二十年来引起了人们的广泛关注，并在混沌分形理论的探讨和混沌分形工程应用等方面均取得了可喜的研究成果。本文对混沌分形理论进行了系统深入的研究，在此研究成果的基础上，对机构学、机器人及全局优化设计主要问题中的混沌分形现象做出了详细的分析，并针对混沌分形在计算运动学、全局优化技术、进化计算等方面的应用做了深入而富有创造性的研究和试验验证，研究面向机构学及全局优化设计中一些一直比较困难的课题。本文的主要内容与创新成果包括以下几个方面：
第一章(绪论)系统回顾了混沌分形的研究历史和发展趋势，详细介绍了混沌分形在机构学及优化设计中的研究进展。在仔细研究了计算运动学、全局优化技术以及进化计算的发展现状的基础上，阐明了课题研究的意义和本文的研究内容。
第二章(混沌分形理论与方法的研究)系统阐述了混沌分形动力学的相关定义和定理，证明了Li-Yorke混沌定义与Devaney混沌定义之间关系和有关命题，研究了Julia点集的混沌特性，证明有关Julia集的关键命题，为混沌分形技术的应用奠定了坚实的理论基础。提出了确定动力系统倍周期分叉点的一种基于遗传算法的计算方法及提出了寻找迭代函数Julia点的方法，为混沌在机构学及优化设计中的应用提供了有效的方法。研究了离散动力系统和连续动力系统的Lyapunov指数，提出了定量识别混沌的新方法：计算最大Lvapunov指数的变分法，为判定混沌是否存在提供了有效方法。
第三章(混沌分形在机构学中的应用研究)将混沌分形技术作为机构学综合问题数值求解的新工具，系统分析了计算运动学各种理论方法的优缺点，研究了计算运动学中的混沌分形现象，通过理论证明和数值试验验证了Newton-Raphson迭代法具有混沌分形特性，提出了求解NR迭代函数Julia点的优化模型及进化规划求解方法，利用非线性离散动力系统在其Julia集出现混沌分形现象的特点，提出了一种基于NR搜索技术的求解机构综合问题全部／部分解的混沌分形方法。并在刚体导引和轨迹再现两大平面机构综合问题上进行了应用研究，比同伦方法找到了更多的有意义的实解。
第四章(混沌分形在无约束全局优化设计中的应用研究)系统地提出了基于混沌分形的无约束全局优化新方法，分析了机械优化设计中各种无约束全局优化方法的优缺点，对牛顿优化方法所构成的非线性离散动力系统的混动分形动力行为进行了深入研究，利用混沌分形理论研究牛顿优化方法对初始
第日页
西南交通大学博士研究生学位论文
点敏感的原因，提出了一种求解牛顿优化迭代函数的Julia点的反函数迭代
方法，发现J。lia集一般具有分形结构，而产生此动力系统的迭代函数在其
Julia集上呈混沌现象，利用牛顿优化方法的混沌分形敏感区域，提出了一种
全局优化的新方法，求解非线性优化问题的全部和/或全局最优解。给出了基
于混沌分形的全局优化新方法的计算步骤，并在函数发生优化综合问题和刚
体导引优化综合问题上进行了应用研究和数值试验，证明了该方法的有效性。
第五章(混沌在约束全局优化设计中的应用研究)提出了基于连续时间
变量的混沌全局优化新方法，证明子贯吐连续系统具有一定的优化能力，同
时证明了通过调整适当的受迫力，惯性耗散系统产生的混沌运动能得到控制，
系统通过混沌运动的方式在众多局部极小点之间迁移，随着受迫力的逐渐消
失，系统最终能收敛到全局最优解，在此理论基础上提出了具有全局优化能
力的基于连续时间变量的混沌优化算法。并研究了应用于冗余度机器人在有
障碍的环境中逆运动学问题优化求解，首次提出了一种避开任意多边形的避
障算法，通过构造避障势函数使该混沌全局优化算法能够处理该约束优化问
题。对平面7自由度操作手的多个仿真试验表明，该优化算法不仅能进行避
障运动而且位置误差非常小。
第六章(混沌进化计算及其在机构学及优化设计中的应用)建立了混沌进
化计算的理论框架，提出了求解复杂工程优化问题的全部/部分极小点的全局
优化算法—混沌进化计算方法。论述了进化计算的理论基础，建立了进化
计算的统一模型;研究了混沌发生器的随机性、遍历性和规律性以及混沌与
进化计算相结合的三种方式。设计了相应的混沌进化算子以及混沌群体嵌入
算子，提出了基于混沌吸引域概念的种群保护策略，适应了极值点分布不均
匀的情况，达到求解全部/大部分极小点的目的。针对机械优化设计中普遍存
在的混合离散变量问题，完整地阐明了利用线性表技术实现离散变量的编码
方法，连续变量按加工精度离散化编码方法，多余码的处理技术的混合个体
表示方法，并用于机械工程优化设计实例求解。并将混沌进化计算应用于机
构运动链同构识别中，建立了机构运动链同构识别全新的优化模型，提出了
用有序编码来描述机构运动链的变换矩阵，设计了有效的混沌有序进化算子，
实例研究及计算机仿真，显示该方法是成功的。
最后，结论部分对本文工作进行了总结，指出了进一步研究的方向。
关键词:混沌;分形;计算运动学;全局优化设计;进化计算
Chaos and Fractals, as the important fields of nonlinear system science, have been paid great attention in recent years, and a great deal of achievements has been made both in the theory and application in engineering. In this dissertation Chaotic and Fractal theory have been studied thoroughly and systematically, and the chaotic and fractal phenomenas of problems in mechanisms, robotics and design optimization have been detailed analyzed, and the application for computational kinematics, design optimization and evolution computation have been investigated in a creative way. The researches are all oriented to the difficulty problems in mechanisms and global optimization. The main researches and contributions of this dissertation are as follows:Definitions and theorems pertaining to the chaotic and fractal dynamics are investigated systematically. The relation between Li-Yorke chaotic and Devaney chaotic definitions and the related propositions are demonstrated and proved, and the chaotic characteristics of Julia set are studied and key propositions of Julia set are proved. Two new methods toward the chaos are proposed which provides an effective tool for the application of chaos to mechanisms and design optimization: method of computing the bifurcation accurately based on genetic algorithms and method of finding the Julia set point of iteration function. Lyapunov exponent for discrete and continuous dynamic systems are investigated, and an effective criteria for determining chaos based on computing the largest Lyapunov exponent using variation method is proposed.Chaotic and fractal technique is used as a tool for solving problems of mechanism synthesis for the first time, and the advantages and disadvantages of several computational kinematics theories are analyzed. The phenomena of chaos and fractals in computational kinematics are investigated, and the phenomena that chaos and fractal does exist in Newton-Raphson iterative method are proved both in theory and numerical simulation. The optimization model for finding Julia set point of Newton-Raphson iteration function is proposed and solved by evolutionary programming. Based on that Julia set is the boundaries of basins of attractions(roots) a novel method for obtaining all solutions of system of nonlinear equations arising from mechanism synthesis problem by utilizing sensitive fractal areas to locate the Julia set point is proposed, and planar rigid body guidance with five precision positions and planar path generation with nine precision points are solved by the method, and more meaningful solutions have been found compared with that by Homotopy method.An unconstrained optimization method for global solution based on chaos and fractal is proposed for the first time. The chaotic and fractal dynamic characteristics of discrete nonlinear system in Newton technique optimization are studied deeply, and reasons that sensitivity of Newton technique optimization heavily dependents on the initial guess point are investigated by means of chaos
and fractal theory. Based on that Julia set is the boundaries of basins of attractions (optimal) a new global optimization method of finding all local optima in nonlinear optimization problem is proposed by utilizing sensitive fractal areas to locate the Julia set point. The developed technique uses an important feature of fractals to preserve shape of basins of attraction(optima) on infinitely small scales. The method has been applied to the approximate synthesis of planar function generation and planar rigid body guidance effectively.The chaotic global optimization method based on continuous time variables is also proposed for the first time. That the continuous inertial system has optimization ability is proved. By adjusting compelling force adequately the chaos motion generated by inertial dissipative system is controlled to let the system migrating among the local minima and finally converging to global minimum. The proposed method is aplicated to the problem of point-to-point motion of redundant robot manipulators working in the
第一章(绪论)系统回顾了混沌分形的研究历史和发展趋势，详细介绍了混沌分形在机构学及优化设计中的研究进展。在仔细研究了计算运动学、全局优化技术以及进化计算的发展现状的基础上，阐明了课题研究的意义和本文的研究内容。
第二章(混沌分形理论与方法的研究)系统阐述了混沌分形动力学的相关定义和定理，证明了Li-Yorke混沌定义与Devaney混沌定义之间关系和有关命题，研究了Julia点集的混沌特性，证明有关Julia集的关键命题，为混沌分形技术的应用奠定了坚实的理论基础。提出了确定动力系统倍周期分叉点的一种基于遗传算法的计算方法及提出了寻找迭代函数Julia点的方法，为混沌在机构学及优化设计中的应用提供了有效的方法。研究了离散动力系统和连续动力系统的Lyapunov指数，提出了定量识别混沌的新方法：计算最大Lvapunov指数的变分法，为判定混沌是否存在提供了有效方法。
第三章(混沌分形在机构学中的应用研究)将混沌分形技术作为机构学综合问题数值求解的新工具，系统分析了计算运动学各种理论方法的优缺点，研究了计算运动学中的混沌分形现象，通过理论证明和数值试验验证了Newton-Raphson迭代法具有混沌分形特性，提出了求解NR迭代函数Julia点的优化模型及进化规划求解方法，利用非线性离散动力系统在其Julia集出现混沌分形现象的特点，提出了一种基于NR搜索技术的求解机构综合问题全部／部分解的混沌分形方法。并在刚体导引和轨迹再现两大平面机构综合问题上进行了应用研究，比同伦方法找到了更多的有意义的实解。
第四章(混沌分形在无约束全局优化设计中的应用研究)系统地提出了基于混沌分形的无约束全局优化新方法，分析了机械优化设计中各种无约束全局优化方法的优缺点，对牛顿优化方法所构成的非线性离散动力系统的混动分形动力行为进行了深入研究，利用混沌分形理论研究牛顿优化方法对初始
第日页
西南交通大学博士研究生学位论文
点敏感的原因，提出了一种求解牛顿优化迭代函数的Julia点的反函数迭代
方法，发现J。lia集一般具有分形结构，而产生此动力系统的迭代函数在其
Julia集上呈混沌现象，利用牛顿优化方法的混沌分形敏感区域，提出了一种
全局优化的新方法，求解非线性优化问题的全部和/或全局最优解。给出了基
于混沌分形的全局优化新方法的计算步骤，并在函数发生优化综合问题和刚
体导引优化综合问题上进行了应用研究和数值试验，证明了该方法的有效性。
第五章(混沌在约束全局优化设计中的应用研究)提出了基于连续时间
变量的混沌全局优化新方法，证明子贯吐连续系统具有一定的优化能力，同
时证明了通过调整适当的受迫力，惯性耗散系统产生的混沌运动能得到控制，
系统通过混沌运动的方式在众多局部极小点之间迁移，随着受迫力的逐渐消
失，系统最终能收敛到全局最优解，在此理论基础上提出了具有全局优化能
力的基于连续时间变量的混沌优化算法。并研究了应用于冗余度机器人在有
障碍的环境中逆运动学问题优化求解，首次提出了一种避开任意多边形的避
障算法，通过构造避障势函数使该混沌全局优化算法能够处理该约束优化问
题。对平面7自由度操作手的多个仿真试验表明，该优化算法不仅能进行避
障运动而且位置误差非常小。
第六章(混沌进化计算及其在机构学及优化设计中的应用)建立了混沌进
化计算的理论框架，提出了求解复杂工程优化问题的全部/部分极小点的全局
优化算法—混沌进化计算方法。论述了进化计算的理论基础，建立了进化
计算的统一模型;研究了混沌发生器的随机性、遍历性和规律性以及混沌与
进化计算相结合的三种方式。设计了相应的混沌进化算子以及混沌群体嵌入
算子，提出了基于混沌吸引域概念的种群保护策略，适应了极值点分布不均
匀的情况，达到求解全部/大部分极小点的目的。针对机械优化设计中普遍存
在的混合离散变量问题，完整地阐明了利用线性表技术实现离散变量的编码
方法，连续变量按加工精度离散化编码方法，多余码的处理技术的混合个体
表示方法，并用于机械工程优化设计实例求解。并将混沌进化计算应用于机
构运动链同构识别中，建立了机构运动链同构识别全新的优化模型，提出了
用有序编码来描述机构运动链的变换矩阵，设计了有效的混沌有序进化算子，
实例研究及计算机仿真，显示该方法是成功的。
最后，结论部分对本文工作进行了总结，指出了进一步研究的方向。
关键词:混沌;分形;计算运动学;全局优化设计;进化计算
Chaos and Fractals, as the important fields of nonlinear system science, have been paid great attention in recent years, and a great deal of achievements has been made both in the theory and application in engineering. In this dissertation Chaotic and Fractal theory have been studied thoroughly and systematically, and the chaotic and fractal phenomenas of problems in mechanisms, robotics and design optimization have been detailed analyzed, and the application for computational kinematics, design optimization and evolution computation have been investigated in a creative way. The researches are all oriented to the difficulty problems in mechanisms and global optimization. The main researches and contributions of this dissertation are as follows:Definitions and theorems pertaining to the chaotic and fractal dynamics are investigated systematically. The relation between Li-Yorke chaotic and Devaney chaotic definitions and the related propositions are demonstrated and proved, and the chaotic characteristics of Julia set are studied and key propositions of Julia set are proved. Two new methods toward the chaos are proposed which provides an effective tool for the application of chaos to mechanisms and design optimization: method of computing the bifurcation accurately based on genetic algorithms and method of finding the Julia set point of iteration function. Lyapunov exponent for discrete and continuous dynamic systems are investigated, and an effective criteria for determining chaos based on computing the largest Lyapunov exponent using variation method is proposed.Chaotic and fractal technique is used as a tool for solving problems of mechanism synthesis for the first time, and the advantages and disadvantages of several computational kinematics theories are analyzed. The phenomena of chaos and fractals in computational kinematics are investigated, and the phenomena that chaos and fractal does exist in Newton-Raphson iterative method are proved both in theory and numerical simulation. The optimization model for finding Julia set point of Newton-Raphson iteration function is proposed and solved by evolutionary programming. Based on that Julia set is the boundaries of basins of attractions(roots) a novel method for obtaining all solutions of system of nonlinear equations arising from mechanism synthesis problem by utilizing sensitive fractal areas to locate the Julia set point is proposed, and planar rigid body guidance with five precision positions and planar path generation with nine precision points are solved by the method, and more meaningful solutions have been found compared with that by Homotopy method.An unconstrained optimization method for global solution based on chaos and fractal is proposed for the first time. The chaotic and fractal dynamic characteristics of discrete nonlinear system in Newton technique optimization are studied deeply, and reasons that sensitivity of Newton technique optimization heavily dependents on the initial guess point are investigated by means of chaos
and fractal theory. Based on that Julia set is the boundaries of basins of attractions (optimal) a new global optimization method of finding all local optima in nonlinear optimization problem is proposed by utilizing sensitive fractal areas to locate the Julia set point. The developed technique uses an important feature of fractals to preserve shape of basins of attraction(optima) on infinitely small scales. The method has been applied to the approximate synthesis of planar function generation and planar rigid body guidance effectively.The chaotic global optimization method based on continuous time variables is also proposed for the first time. That the continuous inertial system has optimization ability is proved. By adjusting compelling force adequately the chaos motion generated by inertial dissipative system is controlled to let the system migrating among the local minima and finally converging to global minimum. The proposed method is aplicated to the problem of point-to-point motion of redundant robot manipulators working in the
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