Forward Displacement Analysis of a Quadratic Planar Parallel Manipulator: 3-RP̱R Parallel Manipulator With Similar Triangular Platforms ¹

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) of a quadratic parallel manipulator: $3-RP̱R$ planar parallel manipulator with similar triangular platforms. Although it has been revealed numerically elsewhere that for this parallel manipulator, the four solutions to the FDA fall, respectively, into its four singularity-free regions (in its workspace), it is unclear if there exists a one-to-one correspondence between the four formulas, each producing one solution to the FDA, and the four singularity-free regions. Using an algebraic approach, this paper will prove that such a one-to-one correspondence exists. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified.