Application of Grassmann-Cayley Algebra to Geometrical Interpretation of Parallel Robot Singularities

The aim of this paper is two-fold: first, it provides an overview of the implementation of Grassmann-Cayley algebra to the study of singularities of parallel robots; and, second, this algebra is utilized to solve the singularity of a general class of Gough-Stewart platforms (GSPs). The Grassmann-Cayley algebra has an intuitive way of representing geometric entities and writing them and their incidence algebraically. The singularity analysis is performed using the bracket representation of the Jacobian matrix determinant associated with this algebra. This representation is a coordinate-free one, and for all cases treated and addressed in this paper, it enables the translation of the algebraic expression into a geometrically meaningful statement. The class of GSPs having two pairs of collocated joints, whose singularity is treated in this paper, is one of the more general classes. Their singularity analysis and geometrical interpretation, is presented here, to the best of our knowledge, for the first time.