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• An automatic method is presented to synthesize contracted graphs of planar kinematic chains. • The vertex degree sequence equation of contracted graph is presented. • A compound topologicassl invariant based method is proposed to detect isomorphism. • The important characteristics of similarity and planarity in contracted graphs are discussed. • Non-fractionated contracted graphs with up to seven independent loops are synthesized. The contracted graph reflects the primary topology of kinematic chains (KCs) and the synthesis of contracted graphs is the foundation for the structural synthesis of KCs. This paper presents an automatic method to synthesize contracted graphs of planar non-fractionated KCs. First, according to the range of the number of vertices in a contracted graph, the vertex degree sequence (VDS) equation of contracted graph is established to derive all possible VDSs. Then, the process of solving the synthesis equation set of contracted graphs is developed, based on which all candidate contracted graphs are generated. Finally, unconnected and fractionated contracted graphs are detected, and a compound topological invariant (CTI) based method is proposed to detect isomorphic contracted graphs. The complete set of non-fractionated contracted graphs with up to seven independent loops are synthesized. All the contracted graphs are classified into planar and non-planar graphs, and similar vertices in each contracted graph are determined, which can improve the efficiency of KC synthesis. [ABSTRACT FROM AUTHOR]

• Method for the similarity recognition and isomorphism identification is proposed. • It effectively detects planar single and multiple joints KCs, etc. • Remarkable efficiency advantages in identifying isomorphism of multibar single-hinged KC. • Isomorphism identification method conforms to the definition of isomorphism in graph theory. Similarity recognition and isomorphism identification (SRII) of kinematic chains (KCs) are key problems facing their innovative regenerative design and configuration synthesis. Although various detection methods have been proposed, most of them are based on topological characteristics or some codings, which are only suitable for single type KC and error identification is generated in some cases. In this study, a set of SRII methods is proposed based on the graph theory definition of similarity and isomorphism. These are appropriate for planar single and multiple joints KCs, planetary gear trains (PGTs), contracted graphs, and multicolor graphs. The computational complexity and effectiveness of the algorithm are assessed on 10-bar, 15-bar, 21-bar, 28-bar, and 42-bar single-joint KCs, 4-bar tricolor topological graphs, and 8-bar contracted graphs. All the results are in agreement with those in the cited literature, and the advantage in isomorphism identification efficiency of the multibar single-hinged KC is evident. Finally, SRII are performed for all 8-bar one-degree-of-freedom (1-DOF) and 9-bar 2-DOF planar single-hinged KCs, all 8-bar 1-DOF KCs with two multiple joints, and all 6-bar 1-DOF rotation graphs through identification program, with the aim to provide a reference for the application of the algorithm. [ABSTRACT FROM AUTHOR]

• A synthesis method for eliminating isomorphism identification based on graph similarity is proposed. • Similar edges are further divided into two types, i.e., CESE and IESE. • The internal relations between similar edges and isomorphism are revealed. • Kinematic chains with up to seven loops and three DOFs are exposed. • All the synthesis process can be realized automatically by the computer. A novel method for eliminating isomorphism identification is proposed to improve synthesis efficiency and synthesize planar nonfractionated kinematic chain (KCs) automatically. This method is based on the vertex insertion of contracted graphs. First, similar edges of contracted graphs are divided into groups, in which the similar edges are found and their characteristic matrices are calculated. The edge types are divided based on whether or not isomerism occurs after vertices are inserted. Then, the vertices are inserted into contracted graphs according to the edge condition. In this process, all isomerism caused by the location and number of inserted vertices is reserved, and the property change of similar edges is checked. Lastly, the rigid subchains of remaining isomerism are distinguished. Contracted graphs with four independent loops and some of their inserted vertices are presented in appendix. A complete set of nonfractionated KCs with up to seven independent loops and three degrees of freedom is also provided. The veracity and efficiency of the method are confirmed by conducting a comparative analysis between this synthesis and other literature results. [ABSTRACT FROM AUTHOR]