Computer Aided Detection of Graph Equivalence Among Multiple Jointed Kinematic Chains and Their Links

Abstract

Graph isomorphism problem essentially involves the comparison of the structures of a given pair of graphs for similarity. It is a research topic that originated from “graph theory”. This problem has been receiving wide attention mainly because of its practical importance. As kinematic chains can be drawn in different ways, it is difficult to detect similarity in kinematic structures by visual inspection. In the structural synthesis and analysis of kinematic chains the isomorphism problem in chains merits special attention since all the existing methods generate a number of isomorphic chains, and the structural analysis of kinematic chains can be expressed as one of an isomorphism problem. Many methods are available for a kinematician in case of simple jointed kinematic chains but there are very few methods in case of multiple jointed kinematic chains. In fact, the most tedious task in structural synthesis of multiple jointed kinematic chains is that of comparison for isomorphism since it does not render itself easily for adoption of adjacency (0, 1) matrices. Using the loop technique a method is proposed in this paper which is based on the disposition of ‘edge’ (here in after called ‘links’) and ‘vertices’ (here in after called ‘joints’) in various loop formations. A new structural invariant called “Multiple Jointed Chain loop string” is developed for a kinematic chain with multiple joints, which takes into account the value of the joints. Another invariant called the “Coincident Link String” is developed which is a by-product of the same method to identify distinct mechanisms for a given kinematic chain.