Application of the hamming number technique to detect isomorphism among kinematic chains and inversions

Abstract Many methods are available to the kinematician to detect isomorphism among chains and among inversions but each has its own shortcomings. In the present paper a novel approach, which is both reliable and simple, is presented. Use is made of the Hamming number, a concept borrowed from digital communication theory. The connectivity matrix of the various links, a matrix of zeros and ones, is first formed and the Hamming number matrix is computed. The link Hamming string—which is defined as the string obtained by concatenating the link Hamming number and the frequency of individual Hamming numbers in that row—is then formed. Finally, the chain Hamming string, defined as the string obtained by the concatenation of the chain Hamming number and the link Hamming string is an excellent test for isomorphism among chains. Also, the link Hamming string of every link together with those of its neighbours is an excellent test for isomorphism among the inversions of a given chain. These twin claims have been verified on a computer for all six-, eight- and ten-bar chains with one degree of freedom as well as ten-bar chains with three-degrees of freedom. It is felt that the greatest advantage of this method is that the chain Hamming string reveals at a glance, without much additional computation, how many inversions are possible out of a given chain.