Various types of RC//RC――-like linkages and the discontinuously movable Koenigs joint

Equivalent open chains generating the same motion as RC dyad with parallel axes are introduced based on equivalencies stemming from the algebraic structure of Lie group of the set of Euclidean displacements. These equivalent chains can be used to replace the RC limbs to deduce a family of RC//RC――-like linkages. Cylindrical translation and exceptional mobility of 1-DoF translational linkages having RC limbs or equivalencies are revealed using the Lie-group property of the rigid-body displacement set. The intersections of two circular cylinders offer classification of various types of RC//RC―― linkages and the like. The discontinuous mobility of motion bifurcation in special RC//RC――-like linkages is disclosed. General cases for the same phenomenon occur when an offset of translation displaces two RC subchains or equivalencies. At a singular posture of bifurcation, the coupler-bar locally has 2-DoF of infinitesimal translation and two working modes arise at the position of possible double point on the bicylindrical curve generated by two RC open sub-chains or equivalencies. Finally, the special case of Koenigs joint is described for the potential application in machinery.