On the Singularity Surface of Planar 3-RPR Parallel Mechanisms

This paper investigates the algebraic nature of the singularity surface of 3-RPR planar parallel mechanisms. An old result by Noether shows that the singularity surface in the kinematic image space is rational and hence must consist of a single component. This property implies that if a box can be defined in the workspace of the mechanism such that its faces, edges, and vertices do not contain a singular pose, then there are no singularities inside the box, which is a very useful result in a context of design. The kinematic mapping is first recalled and the equation of the singularity surface is derived in two different ways. The geometric properties of this surface that are essential to derive the rational parametrization are then discussed. Finally, two algorithms to derive parametric representations of the surface are given, which completes the proof.