Singularity Analysis of Lower-Mobility Parallel Manipulators Using Grassmann-Cayley Algebra

TL;DR

This paper presents a geometric methodology using Grassmann-Cayley Algebra to analyze singularities in lower-mobility parallel manipulators, focusing on the dependency of Plücker vectors in the inverse Jacobian matrix.

Contribution

It introduces a novel geometric approach to singularity analysis of manipulators with complex leg configurations, expanding the analytical tools available for such systems.

Findings

Derived singularity conditions in vector form using Grassmann-Cayley Algebra

Applied the methodology to analyze four different manipulators

Provided insights into the dependency of Plücker vectors for singularity detection

Abstract

This paper introduces a methodology to analyze geometrically the singularities of manipulators, of which legs apply both actuation forces and constraint moments to their moving platform. Lower-mobility parallel manipulators and parallel manipulators, of which some legs do not have any spherical joint, are such manipulators. The geometric conditions associated with the dependency of six Pl\"ucker vectors of finite lines or lines at infinity constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley Algebra. Accordingly, the singularity conditions are obtained in vector form. This study is illustrated with the singularity analysis of four manipulators.