机构拓扑学理论的基本思想与数学方法——从方法论角度回顾几种原创性理论与方法

doi:10.3901/JME.2020.03.001

摘要/Abstract

摘要： 机构拓扑学的核心内容包括：基于非逻辑思维提出的基本概念（拓扑结构、运动特征等），由基本概念导出的机构拓扑学基本公式，以及由逻辑推理提出的拓扑综合方法。基本概念及其数学表示，是引入/建立机构拓扑学可运算数学方法的依据（对螺旋理论与线性变换两种方法，是引入了线性运算方法；对位移子群方法，是建立了非线性符号运算方法；对方位特征集方法，是建立了线性符号运算方法）；机构拓扑学的基本概念、基本公式和拓扑综合方法的主要特点具有对应性（由于螺旋理论与线性变换的线性运算与运动位置有关，又与定坐标系有关，结构综合得到瞬时机构，需进行非瞬时性判定；由于位移子群的非线性符号运算与运动位置无关，可得到非瞬时机构，但与定坐标系有关，且未能涵盖子流形机构；由于方位特征集的线性符号运算与运动位置无关，可得到非瞬时机构，又由于与定坐标系无关，得到机构存在的几何条件具有一般性）。总之，基于螺旋理论、线性变换与位移子群的三种方法都与定坐标系有关，属于代数学方法，而基于方位特征集的方法与定坐标系无关，属于内蕴几何学方法。因此，由非逻辑思维提出的基本概念及其数学表示是建立原创性机构拓扑学系统理论与方法的基石。

关键词: 机构拓扑学, 拓扑结构, 运动特征, 基本方程, 拓扑综合, 机构组成原理

Abstract: The essential contents of mechanism topology theory include some basic concepts proposed based on non-logical thinking, basic topological equations established based on these basic concepts and corresponding topological synthesis method proposed based on logical reasoning. Basic concepts and their mathematical representations are the bases for establishment of a computable mathematical method for mechanism topology (e.g. linear operations are introduced for screw theory based method and linear transformation method, non-linear symbolic operations are introduced for displacement subgroup method, linear symbolic operations are introduced for position and orientation set method) and the key factors to differentiate the performance of different topological synthesis methods (e.g. instantaneous mechanisms may be obtained but it is necessary to judge whether the obtained mechanism is an instantaneous one for screw theory based method and the linear transformation method since their linear operations are dependent on the mechanism's motion position, while sub-manifold mechanisms are not covered for displacement subgroup method even though non-instantaneous mechanisms are obtained since its non-linear symbolic operations are independent of the mechanism's motion position, non-instantaneous mechanisms are obtained for position and orientation set method since its linear symbolic operations are independent of the mechanism's motion position). The screw theory based method, the linear transformation method and the displacement subgroup method are algebraic methods since their operations are dependent on the fixed coordinate system. While the position and orientation set method is an intrinsic geometric method since its operations are independent on the fixed coordinate system. In a word, the research shows that the basic concepts proposed based on non-logical thinking and their mathematical representations are the cornerstones for establishment of original mechanism topology theories and methods.

Key words: mechanism topology, topological structure, kinematic characteristics, basic equations, topological synthesis, mechanism composition theory

中图分类号:

TG156

杨廷力, 沈惠平, 刘安心, 杭鲁滨. 机构拓扑学理论的基本思想与数学方法——从方法论角度回顾几种原创性理论与方法[J]. 机械工程学报, 2020, 56(3): 1-15.

YANG Tingli, SHEN Huipng, LIU Anxin, HANG Lubin. Basic Ideas and Mathematical Methods of Mechanism Topology Theory——Review of Several Original Mechanism Topology Theories in a Methodological Perspective[J]. Journal of Mechanical Engineering, 2020, 56(3): 1-15.

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