Basic Ideas and Mathematical Methods of Mechanism Topology Theory --Review of Several Original Mechanism Topology Theories in a Methodological Perspective

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.