Topology of the singularities of 3-RPR planar parallel robots

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. also shows path two symmetrical solutions if appropriate condition satisfied. This will be automatically satisfied for exactly one pair four real to the kinematics. The topology kinematic singularities these robots described. prove complement singularity space in domain map such consists three connected components. An analysis special provided, i.e. when anchor points moving platform and fixed have same cross-ratios. For this case we show that, components, can without crossing surface absolute value determinant Jacobian evaluates number any topological platforms based on fibers natural SE(2)?S1 restricted set map, conic fibration. In fact or both triangular. use similar methodology as triangles has result was already but here different approach followed.