Singular surfaces and cusps in symmetric planar 3-RPR manipulators

TL;DR

This paper analyzes a class of symmetric 3-RPR manipulators, providing geometric characterizations, singularity descriptions, and methods for motion planning based on cusp edges and assembly mode sorting.

Contribution

It introduces a coordinate system tailored to the manipulators' geometry, simplifying the analysis of singularities and enabling effective motion planning strategies.

Findings

Explicit descriptions of singularities and cusp edges.

A method for sorting assembly modes.

Enhanced motion planning in joint space.

Abstract

We study in this paper a class of 3-RPR manipulators for which the direct kinematic problem (DKP) is split into a cubic problem followed by a quadratic one. These manipulators are geometrically characterized by the fact that the moving triangle is the image of the base triangle by an indirect isometry. We introduce a specific coordinate system adapted to this geometric feature and which is also well adapted to the splitting of the DKP. This allows us to obtain easily precise descriptions of the singularities and of the cusp edges. These latter second order singularities are important for nonsingular assembly mode changing. We show how to sort assembly modes and use this sorting for motion planning in the joint space.