  We study in this paper a class of 3-RPR manipulators for which the direct
kinematic problem (DKP) is split into a cubic problem followed by a quadratic
one. These manipulators are geometrically characterized by the fact that the
moving triangle is the image of the base triangle by an indirect isometry. We
introduce a specific coordinate system adapted to this geometric feature and
which is also well adapted to the splitting of the DKP. This allows us to
obtain easily precise descriptions of the singularities and of the cusp edges.
These latter second order singularities are important for nonsingular assembly
mode changing. We show how to sort assembly modes and use this sorting for
motion planning in the joint space.
