Analytical Determination of the Longest Cylinder Free of Gain-type Singularities Inside the Workspace of a$$3$$ -RPS Spatial Manipulator

The knowledge of singularity-free regions (SFRs) inside the workspace of a manipulator is helpful in its path-planning and design. To identify such regions analytically, the singularity manifold needs to be determined first. This paper presents the derivation of the gain-type singularity manifold in the task-space of the -RPS manipulator, which is further utilised to compute its SFR, in the form of a singularity-free cylinder (SFC), which is free of gain-type singularities. The problem of identifying the longest SFC for a given constant radius and a chosen base circle is posed as a constrained optimisation problem, which reduces to a 21-degree univariate polynomial in the length of the SFC. The formulation is illustrated via a numerical example.