On Some Notable Singularities of ${3{\text {-}}{} { R\underline{P}}R}$ and ${3{\text {-}}{} { \underline{R}RR}}$ PPRMs

Abstract

This paper highlights the existence of some notable type-II singular configurations for certain planar parallel robotic manipulators (PPRMs). These singularities are characterized by intrinsic geometric conditions of alignment or coincidence between geometric entities of the fixed base, the mobile platform and the limbs. Thus, in the general case of the ${3{\text{- }}R\underline{P}R}$ manipulator, a set of 6 such singular configurations can be identified for each orientation of the mobile platform. Moreover, for 6 particular orientations of the mobile platform, a set of positions of the end-effector, defined by two concurrent lines, can be identified as notable degenerated singularity curves. On another side, in the general case of the ${3{\text{- }}\underline{R}RR}$ manipulator, a set of 24 curves in the 3-dimensional operational space (x, y, $\beta$) can be identified as singular poses. All these singularities are easy to determine by means of simple geometric graphical constructions. In this paper, we try to exploit the existence of such particular singularities for kinematic analysis and design of PPRMs. For instance, we can construct the singularity surface of the ${3{\text{- }}R\underline{P}R}$ manipulator by using a pure graphical approach and without any need of algebraic or analytic formulations.