Given a general 3-RPR planar parallel robot with linear platforms, it is proven that there are no singularity-free paths between non-symmetrical direct kinematics solutions. We provide an alternative proof of this. We also provide a proof that shows that there is a singularity-free path between two symmetrical solutions if an appropriate condition is satisfied. This condition will be automatically satisfied for exactly one symmetrical pair of solutions if there are four real solutions to the direct kinematics. The topology of the kinematic singularities of these robots is described. We prove that the complement of the singularity space in the domain of the kinematic map for such robots consists of three connected components. An analysis of special robots is also provided, i.e. when the anchor points of the moving platform and the fixed platform have the same cross-ratios. For this case we show that, the singularity-free space consists of four connected components, no direct kinematics solutions can be connected without crossing the singularity surface and that the absolute value of the determinant of the Jacobian of the kinematic map evaluates to the same number for any of the direct kinematics solutions. The topological analysis of the singularities of 3-RPR planar parallel robots with linear platforms is based on the fibers of the natural map SE(2)→S1 restricted to the singularity set of the kinematic map, and this is a conic fibration. In fact this is also the case when one or both platforms are triangular. We use similar methodology as in the linear platforms case and show that in the case that one or both platforms are triangles the singularity-free space has two connected components. This result was already proven when both platforms are triangles but here a different approach is followed.

Answering connectivity queries in semi-algebraic sets is a longstanding and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying in the same connected component of such a set. In this paper, we push forward the current capabilities of computer algebra to obtain computer-aided proofs of the analysis of the kinematic singularities of various robots used in industry.