On the determination of cusp points of 3-R\underline{P}R parallel manipulators

TL;DR

This paper introduces a rigorous method using discriminant varieties and Gr"obner bases to accurately identify all cusp points in 3-RPR parallel manipulators, enhancing understanding of their kinematic behavior.

Contribution

It proposes a novel, mathematically rigorous methodology to determine all cusp points of 3-RPR manipulators, improving upon previous polynomial-based approaches.

Findings

Validated the methodology on 3-RPR manipulators

Ensured all cusp points are identified

Enhanced understanding of manipulator kinematics

Abstract

This paper investigates the cuspidal configurations of 3-RPR parallel manipulators that may appear on their singular surfaces in the joint space. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a non-singular change of assembly mode. In previous works, the cusp points were calculated in sections of the joint space by solving a 24th-degree polynomial without any proof that this polynomial was the only one that gives all solutions. The purpose of this study is to propose a rigorous methodology to determine the cusp points of 3-R\underline{P}R manipulators and to certify that all cusp points are found. This methodology uses the notion of discriminant varieties and resorts to Gr\"obner bases for the solutions of systems of equations.