Closed-Form Derivation of the Gain-Type Singularity Surface of the 3-RRS Parallel Manipulator

Identification and avoidance of singularities of the gain-type or forward kinematic singularities are critically important from the perspectives of design and operation of parallel manipulators. The mathematical conditions for such singularities are easy to derive, since these commonly reduce to the vanishing of the determinants of certain pose-dependent matrices. The expansion of these determinants, however, is known to be extremely challenging, especially, in their symbolic forms. Consequently, the set of singular points, or the singularity (hyper)-surface, remains unknown for many parallel manipulators commonly in use. In this paper, this problem is solved for the  \(3\) -RRS spatial manipulator, utilising a newly introduced computational tool, namely, the Barik transform. The singularity surface is derived in the closed-form for the generic architecture, for the first time. The method presented can be extended to other parallel manipulators.