Research On Identification Of The Isomorphism And The Topological Symmetry In Planar Kinematic Chains Based On The Topological Index

Mechanical structure is the foundation of the invention and creation about mechanical product.The innovative design of structure determines the innovation of mechanical product.The creative design method of mechanical devices provides a systematic theory for the innovative design of mechanical structure.During this process,the isomorphism of kinematic chains and the topological symmetry of the kinematic chain are one of the key and difficult problems,which are of great significance for speeding up the whole design process.In view of the fact that the graph theory model of planar kinematic chain is similar to the molecular model of organic chemistry to a certain extent,in the field of chemistry,molecular topological index is mostly used to screen out the different structures of the same molecule,namely isomers.Therefore,this idea is introduced,the isomorphism and the topological symmetry of planar kinematic chains are studied,and three topological indices of kinematic chains suitable for mechanical field are proposed to solve above two problems effectively in this paper.Firstly,the composition of kinematic chains and topological graphs of various kinematic chains used in this paper are briefly introduced,and the similarities between planar kinematic chains and organic chemical molecules are analyzed.According to the similarities,the bond value between pairs,the connectivity and topological radius of links,the connectivity matrix and the bond matrix of the corresponding planar kinematic chain are defined.At the same time,depending on the topological theory of mechanism,the layer matrix of the planar kinematic chain is elaborated,and a new weighted link adjacency matrix is proposed.Because of the particularity of the bond value defined in this paper,the connectivity matrix and the bond matrix can be directly derived from the adjacency matrix and layer matrix of planar kinematic chain,which is different from the connectivity matrix and bond matrix of organic chemical molecules.Then,a topological index suitable for the isomorphism detection of planar kinematic chains is proposed,i.e.the KC-EAID（The extended adjacency identification index for the isomorphism of the kinematic chains）.If KC-EAID of two planar kinematic chains is the same,then they are isomorphic,otherwise they are not.To obtain high recognition ability,this index introduces the peripheral connection information of links.These information participate in the whole KC-EAID generation process in the form of link weight value and pair weight value.This is more representative of the whole planar kinematic chain structure than the traditional weight value which only considers the single factor of pair or link.The effectiveness of this method has been proved by 10-link 1 degree of free（DOF）planar simple joint kinematic chains,8-link 2-DOF planar kinematic chains with 2-multiple joints and6-link 1-DOF planar planetary gear trains.Finally,the topological symmetries of links and pairs are defined from the perspective of isomorphism,and two topological indices are proposed to identify the topological symmetry of links and pairs in a kinematic chain,namely,the SKC-EAID_ii（The extended adjacency identification index for the symmetry of links in a kinematic chain）representing each link and SKC-EAID_ij（The extended adjacency identification index for the symmetry of pairs in a kinematic chain）representing each pair.If the topological index of two links（Pairs）is the same,then they are topological symmetric links（Pairs）,otherwise they are not.Because the KC-EAID used for isomorphism identification already contains the connection information of each link and pair,the above two topological indices can be easily obtained from the generation process of KC-EAID,which improves the efficiency of mechanical creative design to a certain extent.The topological indices proposed in this paper are integrated into the same operation flow,and this method is simple,efficient and convenient for computer programming.It provides a new powerful tool for identifying isomorphism and topological symmetry in planar kinematic chains.