Squared Shortest Path Distance [SSPD] Matrix Approach to Identify Isomorphic and Non- Kinematic Chains

Abstract:- Isomorphism identification of kinematic chain is one challenging problem in the field of mechanism. This paper attempts to solve the problem of isomorphism among kinematic chains with the help of squared shortest path distance [SSPD] matrix. In this method the given KC’s are represented in the form of squared shortest path distance matrix [SSPD]. The sum of all elements of [SSPD] matrix is considered as an invariant of a kinematic chain which may used to detect isomorphism. With the help of these invariant/identification code the isomorphism among the kinematic chains are identified. No counterexample has been found. The proposed method is efficient and accurate and only one [SSPD] matrix for a given kinematic chain is sufficient to identify isomorphism. This method is examined for one degree of freedom (1-DOF), 6, 8, 10 links planar kinematic chains and 9 links 2-DOF planar kinematic chains.

Keywords: KC, [SSPD]

INTRODUCTION

In the structural synthesis of the kinematic chains One of the important areas is to develop all possible mechanisms derived from a given kinematic chain at the development phase of conceptual design of Structural synthesis of the kinematic chain and mechanism. During the process of enumeration of kinematic chains there is some chance of creating duplicate kinematic chain because of method adopted are not reliable to detect same which leads to isomorphism among the kinematic chains and that is the disease of kinematic chains which must have to eliminate during the enumeration before the development of distinct mechanism so that the designer has the liberty to select the best or optimum mechanism depending upon the requirement. As mentioned before In the course of enumeration of kinematic chains, duplication or isomorphism may be possible. So for the recognition of isomorphism, the researchers have proposed many methods in recent years. The methods proposed so far are based on an adjacency matrix [1] distance matrix [2,3] to determine the structurally distinct mechanisms of a kinematic chain; the flow matrix method [4], and the row sum of extended adjacency matrix methods [5,6] are used. Minimum code [7], characteristic polynomial of a matrix [8], identification code [9], link path code [10], path matrices [11], a Multivalued Neural Network approach [12], a mixed isomorphism approach [13], Hamming value [14], an artificial neural network approach [15], the theory of finite symmetry groups [16,17], the representation set of links by Vijayananda [18], Interactive Weighted Distance Approach [19], are used to characterize the kinematic chains. Among of these methods either have a lack of uniqueness or very time consuming. Hence, there is a need to develop an optimized method to detect isomorphism in kinematic chains
Read More