Hidden cusps

TL;DR

This paper explores the nature of cusps in singularities of planar 2-RPR-PR parallel manipulators, demonstrating that non-singular assembly mode changes involve cusp points and are generally associated with specific stable singularity types.

Contribution

It clarifies the conditions under which cusps occur in manipulator singularities and confirms that non-singular assembly mode changes are typically achieved by encircling cusp points.

Findings

Non-generic change of assembly mode involves cusps under perturbation

Two stable singularity types are identified: complex square and quarto mappings

Non-singular solution changes generally involve encircling cusp points

Abstract

This paper investigates a situation pointed out in a recent paper, in which a non-singular change of assembly mode of a planar 2-RPR-PR parallel manipulator was realized by encircling a point of multiplicity 4. It is shown that this situation is, in fact, a non-generic one and gives rise to cusps under a small perturbation. Furthermore , we show that, for a large class of singularities of multiplicity 4, there are only two types of stable singularities occurring in a small perturbation: these two types are given by the complex square mapping and the quarto mapping. Incidentally , this paper confirms the fact that, generically, a local non-singular change of solution must be accomplished by encircling a cusp point.