Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators

TL;DR

This paper explores two previously unaddressed degeneracy scenarios in the forward kinematics of planar 3-RPR manipulators, revealing new solution behaviors and identifying a novel family of manipulators with simplified kinematic solutions.

Contribution

It introduces and analyzes two new degeneracy cases in the forward kinematics of planar 3-RPR manipulators, including a new family with reduced solution complexity.

Findings

Degeneracy when input joint variables satisfy a specific relationship leads to two solution sets.

Full joint space degeneracy occurs when base and platform triangles are congruent with a 180° rotation.

New family of manipulators with simplified 3rd-degree polynomial kinematic solutions.

Abstract

This paper investigates two situations in which the forward kinematics of planar 3-RPR parallel manipulators degenerates. These situations have not been addressed before. The first degeneracy arises when the three input joint variables r1, r2 and r3 satisfy a certain relationship. This degeneracy yields a double root of the characteristic polynomial in t, which could be erroneously interpreted as two coalesce assembly modes. But, unlike what arises in non-degenerate cases, this double root yields two sets of solutions for the position coordinates (x, y) of the platform. In the second situation, we show that the forward kinematics degenerates over the whole joint space if the base and platform triangles are congruent and the platform triangle is rotated by 180 deg about one of its sides. For these "degenerate" manipulators, which are defined here for the first time, the forward kinematics is reduced to the solution of a 3rd-degree polynomial and a quadratics in sequence. Such manipulators constitute, in turn, a new family of analytic planar manipulators that would be more suitable for industrial applications.