Automatic synthesis of the complete set of contracted graphs for planar kinematic chains with up to seven independent loops

Abstract 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.