Quotient Kinematics Mechanisms: Modeling, Analysis And Synthesis

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.