商联机构：建模、分析与综合
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摘要
随着机械制造工业的飞速发展,高速高精数控加工装备的自主知识产权开发作为对突破西方发达国家技术封锁的关键,近年来受到国家高度重视,并逐步成为国内高校和企业的研究热点。这些装备所需实现的运动指标具有高精度、高速度和高加速度的特点,对装备的机械和机构本体的设计带来了新的挑战。从机构学的角度来看,单一的多自由度机构往往无法再满足日益提高的运动学和动力学指标,相比之下,通过多个低自由度机构(模块)联合运动实现高自由度运动任务的构型方法既可以缩短机构运动链的长度而提高刚度,又可以减少并联模块自身的支链约束而增大平移和旋转工作空间,在上述应用中具有潜在的优势。本论文研究这样一类具有两个独立运动模块的机构,我们称之为商联机构。
为了使商联机构的概念得到广泛接受,并从构型设计上发挥出其性能潜力,首先需要正确认识商联机构中的特殊运动学问题并解决商联机构综合问题。为此本论文提出一套系统的商联机构分析及综合理论框架,研究了商联机构的系统建模、分析和综合问题。商联机构理论的基本思想是在给定目标运动类型的前提下,两个运动模块的运动类型和整体运动类型之间具有类似于商的关系,可以用刚体变换群的商空间进行统一建模,从而引出商联机构的系统分析与综合问题,商联机构也因此得名。
本论文主要解决的问题之一是商联机构的综合问题,由运动类型分解和模块构型综合两方面组成：商联机构的运动类型分解问题是指给定目标运动类型Q,如何系统综合所有的运动类型序对(M1,M2),使由M1和M2构成的商联机构实现Q运动；商联机构的模块构型综合问题是指如何系统地用串联、并联联机构实现M1和M2运动的构型综合问题。
针对上述第一个子问题,本论文运用微分几何和微分拓扑知识,回顾了基于Lie群和微分几何的刚体运动学基础并在此基础上首次提出基于横截子流形的商空间分类方法,并提出了增广规则和缩减规则,建立了商空间的层级关系。商空间分类与层级关系有效解决了对SE(3)的Lie子群和一般子流形进行统一分类的问题,对Lie子群的分类与层级关系进行了重要的推广。增广规则和缩减规则是商联机构综合和商运动类型并联机构综合的核心定理。根据对具有Lie子群运动类型的商联机构的分析,本论文阐述了商联机构综合问题与商运动类型之间的密切联系,并运用增广规则和缩减规则提出了一般商联机构的运动类型分解方法。
针对上述第二个子问题,商联机构作为一种模块化的构型形式,可以包含任意串联、并联和混联的模块,由此引发了作为商联模块的具有商空间运动类型的少自由度并联机构综合等问题。以往,少自由度并联机构的运动类型一般被公认为分为9类,它们对应于刚体变换群SE(3)的9类3-5自由度的子流形。这些子流形的特点是非特征自由度上不存在寄生运动。这类运动类型对应的并联机构的综合问题研究已经基本成熟和完结。相反,商联机构的模块的运动类型出现了商空间的情况,而商空间运动类型并联机构的综合问题从未被前人系统研究。从机构创新角度考虑,商运动类型并联机构的综合理论也可产生大量新颖并联机构,对促进现有并联机构综合理论和相关研究领域的发展。本论文针对具有商运动类型的并联机构,提出了基于增广规则和缩减规则的商运动并联机构系统综合方法,其中间接综合法依赖于商空间横截子流形的选取,而直接综合法进一步突破了对横截子流形的依赖。运用基于商运动类型的并联机构综合方法,我们可以得到大量前所未有的并联机构新型。
商运动并联机构因为其特殊的运动类型,其运动学分析有一定的特殊性。我们通过对一类{SE(3)/PL(Z)}并联机构的举例,研究了商运动并联机构机构综合以外的所有运动学问题：这包括位置正、反解中涉及到寄生运动产生的新问题,以及奇异性分析、灵巧度分析和工作空间参数化和可视化等方面中产生的新问题。
With the rapid development of mechanical manufacturing industry, self-proprietary development of high speed/precision CNC machine tools is becomingincreasingly emphasized by the state, and becomes a hot research area amongdomestic universities and companies. The stringent requirements upon structureand mechanism design of these equipments conflicts with the limited kinematicand dynamic performances a a single mechanism with multiple degrees of freedom(dof) can ofer. In comparison, realizing one motion task with the cooperation ofmultiple mechanisms with lower dofs (we called modules) has the advantage ofhigher stifness due to shorter kinematics chains, and larger workspace due to sim-pler closed loop constraints. The subject of study of this thesis are mechanismswith two cooperating modules, what we call the quotient kinematic machines(QKM).
In order to promote a wide application of the QKM concept, we need tounderstand its kinematics properly and provide necessary tools for systematicsynthesis. Thus the subject of study of this thesis is: systematic modeling,analysis and synthesis of quotient kinematic mechanisms. Utilizing thefacts that the motion type of modules of a QKM often have the form of Liesubgroups and/or quotient spaces of the special Euclidean group, SE(3), weintroduce the analysis and synthesis problem of QKM based on the theory ofquotient spaces.
The main problem to be solved in this thesis is systematic synthesis ofQKMs, which comprises motion type decomposition of QKM and topological syn-thesis of QKM modules: the motion type deocomposition problem of QKM refersto the problem of designating all motion type pairs (M1, M2) such that they forma QKM of a desired motion type Q; the problem of topological synthesis of QKMmodules refers to the systematic synthesis of serial, parallel and hybrid mecha-nisms that realizes each module motion type M1and M2.
For the first aspect, this thesis fully utilizes diferential geometry of SE(3)and some diferential topology to propose for the first time a systematic clas- sification of quotient spaces of SE(3) which is based on the concept of crosssection of a quotient space. Then the core theorem of the thesis, known as theexpansion rule and the reduction rule, is proposed and applied to the elaborationof a hierarchy of quotient spaces of SE(3). The classification and hierarchy ofquotient spaces of SE(3) efectively solves the problem of classification of generalsubmanifolds of SE(3), which serves as an important generalization to that ofLie subgroups of SE(3). The expansion rule and reduction rule is the core theo-rem to motion type decomposition of QKMs and topological synthesis of QKMmodules: through the analysis of Lie subgroup QKMs, we show that the QKMsynthesis problem is closely related to classification and hierarchy of quotientspaces of SE(3), based on which we propose a systematic motion type decompo-sition theory for QKMs with either Lie subgroup motion type or more generallysubmanifold and quotient motion types.
For the second aspect, QKM as a modularized mechanism can have modulesof serial, parallel or even hybrid mechanisms. This has introduced the problem oftopological synthesis of parallel modules with quotient motion types. Previously,low dof parallel mechanism are known to have altogether9types of motion types,in the form of category1submanifolds of SE(3), and widely applied to topo-logical synthesis of parallel mechanisms. Such problems are are widely studiedand is in a terminal stage if not closed. On the other hand, parallel mechanismswith quotient motion types are never formally brought up and systematicallysynthesized. This thesis proposes a systematic topological synthesis method forquotient motion type parallel mechanisms. Two synthesis algorithms are devel-oped: the indirect approach relies on the specification of a cross section of thequotient space in discussion; the direct approach works directly on the quotientspace level.
Quotient parallel mechanism are not only synthetically but also analyticallydistinguished from parallel mechanisms with conventional motion types. Thisthesis studies all aspects of kinematics analysis of a{SE(3)/P L(z)}parallelmechanism toward a better understanding of quotient parallel mechanisms andhopefully a general theory in the future.
为了使商联机构的概念得到广泛接受,并从构型设计上发挥出其性能潜力,首先需要正确认识商联机构中的特殊运动学问题并解决商联机构综合问题。为此本论文提出一套系统的商联机构分析及综合理论框架,研究了商联机构的系统建模、分析和综合问题。商联机构理论的基本思想是在给定目标运动类型的前提下,两个运动模块的运动类型和整体运动类型之间具有类似于商的关系,可以用刚体变换群的商空间进行统一建模,从而引出商联机构的系统分析与综合问题,商联机构也因此得名。
本论文主要解决的问题之一是商联机构的综合问题,由运动类型分解和模块构型综合两方面组成：商联机构的运动类型分解问题是指给定目标运动类型Q,如何系统综合所有的运动类型序对(M1,M2),使由M1和M2构成的商联机构实现Q运动；商联机构的模块构型综合问题是指如何系统地用串联、并联联机构实现M1和M2运动的构型综合问题。
针对上述第一个子问题,本论文运用微分几何和微分拓扑知识,回顾了基于Lie群和微分几何的刚体运动学基础并在此基础上首次提出基于横截子流形的商空间分类方法,并提出了增广规则和缩减规则,建立了商空间的层级关系。商空间分类与层级关系有效解决了对SE(3)的Lie子群和一般子流形进行统一分类的问题,对Lie子群的分类与层级关系进行了重要的推广。增广规则和缩减规则是商联机构综合和商运动类型并联机构综合的核心定理。根据对具有Lie子群运动类型的商联机构的分析,本论文阐述了商联机构综合问题与商运动类型之间的密切联系,并运用增广规则和缩减规则提出了一般商联机构的运动类型分解方法。
针对上述第二个子问题,商联机构作为一种模块化的构型形式,可以包含任意串联、并联和混联的模块,由此引发了作为商联模块的具有商空间运动类型的少自由度并联机构综合等问题。以往,少自由度并联机构的运动类型一般被公认为分为9类,它们对应于刚体变换群SE(3)的9类3-5自由度的子流形。这些子流形的特点是非特征自由度上不存在寄生运动。这类运动类型对应的并联机构的综合问题研究已经基本成熟和完结。相反,商联机构的模块的运动类型出现了商空间的情况,而商空间运动类型并联机构的综合问题从未被前人系统研究。从机构创新角度考虑,商运动类型并联机构的综合理论也可产生大量新颖并联机构,对促进现有并联机构综合理论和相关研究领域的发展。本论文针对具有商运动类型的并联机构,提出了基于增广规则和缩减规则的商运动并联机构系统综合方法,其中间接综合法依赖于商空间横截子流形的选取,而直接综合法进一步突破了对横截子流形的依赖。运用基于商运动类型的并联机构综合方法,我们可以得到大量前所未有的并联机构新型。
商运动并联机构因为其特殊的运动类型,其运动学分析有一定的特殊性。我们通过对一类{SE(3)/PL(Z)}并联机构的举例,研究了商运动并联机构机构综合以外的所有运动学问题：这包括位置正、反解中涉及到寄生运动产生的新问题,以及奇异性分析、灵巧度分析和工作空间参数化和可视化等方面中产生的新问题。
With the rapid development of mechanical manufacturing industry, self-proprietary development of high speed/precision CNC machine tools is becomingincreasingly emphasized by the state, and becomes a hot research area amongdomestic universities and companies. The stringent requirements upon structureand mechanism design of these equipments conflicts with the limited kinematicand dynamic performances a a single mechanism with multiple degrees of freedom(dof) can ofer. In comparison, realizing one motion task with the cooperation ofmultiple mechanisms with lower dofs (we called modules) has the advantage ofhigher stifness due to shorter kinematics chains, and larger workspace due to sim-pler closed loop constraints. The subject of study of this thesis are mechanismswith two cooperating modules, what we call the quotient kinematic machines(QKM).
In order to promote a wide application of the QKM concept, we need tounderstand its kinematics properly and provide necessary tools for systematicsynthesis. Thus the subject of study of this thesis is: systematic modeling,analysis and synthesis of quotient kinematic mechanisms. Utilizing thefacts that the motion type of modules of a QKM often have the form of Liesubgroups and/or quotient spaces of the special Euclidean group, SE(3), weintroduce the analysis and synthesis problem of QKM based on the theory ofquotient spaces.
The main problem to be solved in this thesis is systematic synthesis ofQKMs, which comprises motion type decomposition of QKM and topological syn-thesis of QKM modules: the motion type deocomposition problem of QKM refersto the problem of designating all motion type pairs (M1, M2) such that they forma QKM of a desired motion type Q; the problem of topological synthesis of QKMmodules refers to the systematic synthesis of serial, parallel and hybrid mecha-nisms that realizes each module motion type M1and M2.
For the first aspect, this thesis fully utilizes diferential geometry of SE(3)and some diferential topology to propose for the first time a systematic clas- sification of quotient spaces of SE(3) which is based on the concept of crosssection of a quotient space. Then the core theorem of the thesis, known as theexpansion rule and the reduction rule, is proposed and applied to the elaborationof a hierarchy of quotient spaces of SE(3). The classification and hierarchy ofquotient spaces of SE(3) efectively solves the problem of classification of generalsubmanifolds of SE(3), which serves as an important generalization to that ofLie subgroups of SE(3). The expansion rule and reduction rule is the core theo-rem to motion type decomposition of QKMs and topological synthesis of QKMmodules: through the analysis of Lie subgroup QKMs, we show that the QKMsynthesis problem is closely related to classification and hierarchy of quotientspaces of SE(3), based on which we propose a systematic motion type decompo-sition theory for QKMs with either Lie subgroup motion type or more generallysubmanifold and quotient motion types.
For the second aspect, QKM as a modularized mechanism can have modulesof serial, parallel or even hybrid mechanisms. This has introduced the problem oftopological synthesis of parallel modules with quotient motion types. Previously,low dof parallel mechanism are known to have altogether9types of motion types,in the form of category1submanifolds of SE(3), and widely applied to topo-logical synthesis of parallel mechanisms. Such problems are are widely studiedand is in a terminal stage if not closed. On the other hand, parallel mechanismswith quotient motion types are never formally brought up and systematicallysynthesized. This thesis proposes a systematic topological synthesis method forquotient motion type parallel mechanisms. Two synthesis algorithms are devel-oped: the indirect approach relies on the specification of a cross section of thequotient space in discussion; the direct approach works directly on the quotientspace level.
Quotient parallel mechanism are not only synthetically but also analyticallydistinguished from parallel mechanisms with conventional motion types. Thisthesis studies all aspects of kinematics analysis of a{SE(3)/P L(z)}parallelmechanism toward a better understanding of quotient parallel mechanisms andhopefully a general theory in the future.
引文
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