Analysis of the Topology of the Singularity Set of Planar 3-RPR Parallel Robots

Christoforos Spartalis (1) , Jose Capco (1)

1 Leopold Franzens Universität Innsbruck - University of Innsbruck

Given a general 3-RPR parallel planar robot with linear platforms, [7] proves 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 if there are four real solution to the direct kinematics. The topology of the kinematic singularities of these robots is described. We prove that the singularity-free set of 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 parallel planar robots with linear platforms is based on the fibers of the natural map SE(2)->S1 restricted to the singularity surface 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 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 in [9], but here a different approach is used.

Mots clés: 3-RPR parallel robots; singularities; inverse kinematics; conic fibrations

Domaines: Géométrie algébrique [math.AG]; Robotique [cs.RO]

Christoforos Spartalis, Jose Capco. Analysis of the Topology of the Singularity Set of Planar 3-RPR Parallel Robots. 2022. ⟨hal-03275882v2⟩