A class of analytic planar 3-RPR manipulators is analyzed in this paper.
These manipulators have congruent base and moving platforms and the moving
platform is rotated of 180 deg about an axis in the plane. The forward
kinematics is reduced to the solution of a 3rd-degree polynomial and a
quadratic equation in sequence. The singularities are calculated and plotted in
the joint space. The second-order singularities (cups points), which play an
important role in non-singular change of assembly-mode motions, are also
analyzed.