Identification of Isomorphism in Kinematic Chains by Using the Reduced Graph Matrix

Abstract

—Kinematic chain synthesis normally begins with the generation of a comprehensive list of candidate solutions followed by a time-consuming procedure for isomorph elimination. As a result, the search for isomorphisms in kinematic chains has long attracted the attention of many researchers. Several methods and algorithms have been proposed in the past. Nonetheless, the field still needs fast, efficient and reliable means to prevent duplications across kinematic chains (KC) (i.e., isomorphisms), particularly for configurations with a significant number of bars. Mechanical designers are resorting to kinematic chains and mechanisms with multiple bars to accomplish more complex operations and movements. This complicates the procedure of determining isomorphism. In this paper, we present a simple and efficient method for identifying isomorphisms in kinematic chains by employing a reduced graph matrix, which reduces the adjacency matrix into a compact matrix corresponding to linkages between non-binary bars while implicitly accounting for binary bars. The algorithm’s efficiency and computing complexity are assessed for a number of published situations, including single-joint kinematic chains with 8, 10, 12 bars, and three-complex 13, 15, 28 bars, and lastly 42-bar kinematic chains. This comparison demonstrates the validity and effectiveness of the proposed method.