Necessary and sufficient condition for a generic 3R serial manipulator to be cuspidal

TL;DR

This paper proves that the existence of a cusp point is both necessary and sufficient for any generic 3R serial robot to be cuspidal, extending previous results beyond orthogonal robots and aiding robot design.

Contribution

It generalizes the cusp point condition to all generic 3R robots, providing a comprehensive criterion for cuspidality.

Findings

Cusp point existence is necessary and sufficient for 3R robot cuspidality.

Revisits inverse kinematics and nonsingular posture changes in 3R robots.

Provides a theorem on reduced aspects in generic 3R robots.

Abstract

Cuspidal robots can travel from one inverse kinematic solution to another without meeting a singularity. The name cuspidal was coined based on the existence of a cusp point in the workspace of 3R serial robots. The existence of a cusp point was proved to be a necessary and sufficient condition for orthogonal robots to be cuspidal, but it was not possible to extend this condition to non-orthogonal robots. The goal of this paper is to prove that this condition stands for any generic 3R robot. This result would give the designer more flexibility. In the presented work, the geometrical interpretation of the inverse kinematics of 3R robots is revisited and important observations on the nonsingular change of posture are noted. The paper presents a theorem regarding the existence of reduced aspects in any generic 3R serial robot. Based on these observations and on this theorem, we prove that the existence of a cusp point is a necessary and sufficient condition for any 3R generic robot to be cuspidal.