Advances in the Theory of Planar Curve Cognates

Abstract

Cognate linkages provide the useful property in mechanism design of having the same motion. This paper describes an approach for determining all coupler curve cognates for planar linkages with rotational joints. Although a prior compilation of six-bar cognates due to Dijksman purported to be a complete list, that analysis assumed, without proof, that cognates only arise by permuting link rotations. Our approach eliminates that assumption using arguments concerning the singular foci of the coupler curve to constrain a cognate search and then completing the analysis by solving a precision point problem. This analysis confirms that Dijksman's list for six-bars is comprehensive. As we further demonstrate on an eight-bar and a ten-bar example, the method greatly constrains the set of permutations of link rotations that can possibly lead to cognates, thereby facilitating the discovery of all cognates that arise in that manner. However, for these higher order linkages, the further step of using a precision point test to eliminate the possibility of any other cognates is still beyond our computational capabilities.