Unified pose parametrization for 1T2R parallel manipulators

General-geometry parallel mechanisms (PMs) with one translational and two rotational (1T2R) degrees of freedom (DoF) usually have their end-effector pose trace out a 3-D manifold that cannot be directly parametrized by three physically meaningful coordinates also used for their kinematic analysis. This paper proposes a simple and unified pose parametrization for a general-geometry 1T2R-DoF PM, by identifying an arbitrary pose of the PM with that of a plane attached to its end-effector. The pose of this plane with respect to the same plane at the home configuration is easily represented by a rotation about their intersection line, sometimes referred to as a plane-symmetric motion. The actual pose of the PM decomposes into the product of such a plane-symmetric motion and a subsequent dependent in-plane motion. The proposed parametrization may be applied, for example, to solving the inverse kinematics of 1T2R-DoF PMs.