Compound Topological Invariant Based Method for Detecting Isomorphism in Planar Kinematic Chains

Abstract

An important step in the structural synthesis of kinematic chains (KCs) or mechanisms is the detection of isomorphic structures. Although many detection methods have been proposed, most of them require complex computations and have poor versatility. In this study, a simple detection method is proposed based on a compound topological invariant (CTI), which comprises the fourth power of adjacency matrix and eigenvalues. Besides two complex 15- and 28-link planar simple-joint KCs (PSKCs), the method is tested on the complete atlas of contracted graphs with up to six independent loops, PSKCs with up to 13 links, 8-link 1-degree-of-freedom (DOF) planar multiple-joint KCs (PMKCs), and 6-link 1-DOF planetary gear trains (PGTs). All the results are in agreement with the reported results in the literature. Our method possesses good versatility and has been verified as being reliable and efficient.