Loop Theory Of Kinematic Chains And Its Application To Isomorphism Identification And Atlas Database Establishment

A pivotal issue in mechanical innovation and mechanical conceptual design is the determination of mechanical topological structure. However, the most essential problem of isomorphism identification in the field of mechanical topological structure has remained unsolved over four or five decades, which hampers computer-based automatic mechanical synthesis. Moreover, As regards the synthesized kinematic chains which is so many and hard to identification, there is no theory or method yet in the international academic community that can produce a unique mathematical representation code of kinematic chains; this unique representation should have a one-to-one correspondence with the chains and it should remain the same even when the drawing modes and labeling ways of the topological graphs are changed. For lack of such a code, an atlas database which automatically classify, store, synthesize kinematic chains and in turn display the synthesized chains can not be established. The range of factors hampers the research on mechanical structure from automatic realization by the computer, and mechanical conceptual design is still kept at a relatively low level.In search of solutions to the above-mentioned problems, this paper presents a loop algebra theory and on the basis of the theory establishes a new systematic topological structure theory of kinematic chains. The main contributions of the new theory are as follows.The fundamental mathematic representation of loops of kinematic chains is presented, and 3 mathematic operations on loops are defined, including the corresponding rules and properties. The concept of the basic loop set of kinematic chains and the related theorems are proposed. A theorem on loop relations of kinematic chains is presented and a fairly complete loop algebra theory of kinematic chains is founded.Based on the loop algebra theory, the perimeter topological graph that canonizes the drawing modes of kinematic chains and the canonical perimeter topological graph that canonizes the mathematical representation of the chains are presented. Thereby the canonical adjacency matrix set that describes the kinematic chains uniquely is obtained. A new isomorphism identification method is proposed. It is both effective and efficient and can be encoded, and it applies to all kinematic chains. With this method the extremely difficult problem of isomorphism identification is solved successfully. The algorithm complexity of the new method is compared to that of the methods presently considered the fastest in the world, the Mckay and He methods, and the isomorphism identification method based on the canonical adjacency matrix set is proved highly efficient.The concepts of characteristic perimeter topological graph, characteristic adjacency matrix and synthesis degree sequence of kinematic chains are presented. For the first time a unique digital representation code——the characteristic representation code which applies to all kinematic chains and has a one to one correspondence with the kinematic chain is obtained. The representation code remains unchanged even when the drawing modes and labeling ways of the topological graphs are changed. Thus a one to one correspondence between the topological graph, the characteristic adjacency matrix and the characteristic representation code is established. Based on the characteristic representation code and the VC++ programming language a software system that automatically sketches topological graphs and creates topological graph atlas database of kinematic chains is developed successfully for the first time.Based on the vertex loop theory a structure decomposition theory of kinematic chains is founded, and on the basis of the theory an automatic method of rigid sub-chain detection is proposed. The new method applies to both planar and non-planar kinematic chains. Based on the edge loop theory the rules for selecting driving pairs of kinematic chains with multiple loops and multiple DOFs are given. An optimized selection theory of driving pairs is also researched aiming at simplification of kinematic analysis.