Fractionation in planar kinematic chains: Reconciling enumeration contradictions

Enumeration of kinematic chains satisfying a set of design specifications is still an open problem. Three difficulties are common in the enumeration: isomorphism, degeneration and fractionation. A lot has been said about isomorphism and degeneration but little about fractionation. Some enumeration methods apparently consider fractionated kinematic chains as degenerate kinematic chains and eliminate them. Other methods enumerate fractionated kinematic chains even when a priori it is known that they do not satisfy design specifications. Some of the contradictions in the numbers of kinematic chains are related to fractionation, since the authors in general do not confirm the presence or not of fractionated kinematic chains in their results. It must be remarked that without a method to enumerate purely fractionated kinematic chains it is almost impossible to compare enumeration results with or without fractionation. In this context, a new method of exclusive generation of fractionated kinematic chains is proposed. New results for planar fractionated kinematic chains are obtained and presented in a table. Through these results, it is possible to conclude that most, if not all, contradictions found in the literature are in fact related to fractionation. These results are compared and the contradictions are solved.