Computer Aided Detection of Isomorphism among Kinematic Chains and Mechanisms Using the Concept of Modified Distance

-A computer aided method has been proposed for detecting isomorphism among planar kinematic chains with sample joints using a new invariant called the arranged sequence of modified total distance ranks of all the links (ASMTDRL). This invariant takes into account the degrees of'links, and degrees-of-freedom and types of joints of the kinematic chain. Two computer aided methods have also been developed for identifying distinct mechanisms of a planar kinematic chain with simple joints. I N T R O D U C T I O N Structural analysis and synthesis of kinematic chains is an important aspect of mechanism design. In the early stage of mechanism design, it is helpful to determine all the distinct kinematic chains (in short chains) with the required number of links and degree-of-freedom. One very important problem encountered during structural synthesis of chains is the detection of possible isomorphism among the given chains. Attempts have been made in the past to develop methods for detecting isomorphism among planar chains with simple as well as multiple joints [1-11]. Rao and Varad Raju [12] proposed the hamming number technique to detect isomorphism among chains and inversions. They, however, could derive only 1828 distinct mechanisms from the 230 ten-link, single-degree-of-freedom chains as against 1834 already reported in the literature [4]. Recently, Agrawal et al. [13] developed an invariant called the total distance ranks of all the links arranged in decreasing (or increasing) order, which is based on the concept of distance. This invariant, however, was not found to be sufficient for the test of isomorphism. Further work by the authors on distance concept showed that the discriminating ability of this invariant could be improved by taking into consideration the degrees of links, and degrees-of-freedom and types of joints. In the present paper, a new invariant, called the arranged sequence of modified total distance ranks of all the links, has been developed, taking the above factors into account. Based on this, a computer aided method has been developed for detecting isomorphism among planar chains with simple joints. Two computer aided methods have also been developed for identifying distinct mechanisms of a planar chain with simple joints. DEFINITIONS In the process of developing the new invariant for a chain, some new terms have been introduced and defined [14] as follows: (i) Link-path. A link-path in a chain is an alternating sequence of distinct links and distinct joints, starting and ending with links, such that each joint connects the links preceding and following it in the sequence. For example, the sequence l, a, 2, b, 3, m, l0 is a link-path in the chain shown in Fig. I.