The five types of kinematic chains that generate the planar motion group SE(2) of dimension three, with the prismatic-joint direction always perpendicular with the revolute-joint axis in each chain, have shown their effectiveness and manifested the charm in type synthesis and mechanism analysis in parallel mechanisms. This paper extends the traditional PRP kinematic chain generating the planar motion group SE(2) to a relatively general case, in which one of the prismatic joint-direction is not necessarily perpendicular with the revolute-joint axis, leading to the discovery of a screw motion with a variable pitch in this kinematic chain. Following the extraction of a screw motion from this particular PRP kinematic chain, this paper presents the bifurcated motion in a 3-PUP parallel mechanism by changing the active geometrical constraint in its configuration space, with a Lie group approach and interpretation. The constraint-singularity configuration sets for bifurcation of the 3-PUP parallel mechanism. The paper hence provides a Lie group representation and geometry interpretation for the kinematic equivalence of serial chains and the bifurcated motion of a parallel mechanism.

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