Identification and geometric characterization of Lie triple screw systems and their exponential images

The twist space of a plunging constant-velocity (CV) coupling with intersecting shafts consists, in all configurations, of a planar field of zero-pitch screws. Recently, we reported an important discovery about this screw system: it is closed under two consecutive Lie bracket operations, thus being referred to as a Lie triple system; taking the exponential of all its twists generates the motion manifold of the coupling. In this paper, we first give a geometric characterization of the Lie product and the Lie triple product of a generic screw system. Then, we present a systematic identification of all Lie triple screw systems of se(3), by an approach based on both algebraic Lie group theory and descriptive screw theory. We also derive the exponential motion manifolds of the Lie triple screw systems in dual quaternion representation. Finally, several important applications of Lie triple screw systems in mechanism and machine theory are highlighted in the conclusions.