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

Authors:
Patricia Ben-Horin;Moshe Shoham
Affiliations:
Department of Mechanical Engineering Technion-IsraelInstitute of Technology Technion City, Haifa 32000 Israel;Department of Mechanical Engineering Technion-IsraelInstitute of Technology Technion City, Haifa 32000 Israel
Venue:
International Journal of Robotics Research
Year:
2009

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Abstract

The aim of this paper is two-fold: first, it provides anoverview of the implementation of Grassmann-Cayley algebra to thestudy of singularities of parallel robots1 and, second, thisalgebra is utilized to solve the singularity of a general class ofGough-Stewart platforms (GSPs). The Grassmann-Cayley algebra has anintuitive way of representing geometric entities and writing themand their incidence algebraically. The singularity analysis isperformed using the bracket representation of the Jacobian matrixdeterminant associated with this algebra. This representation is acoordinate-free one, and for all cases treated and addressed inthis paper, it enables the translation of the algebraic expressioninto a geometrically meaningful statement. The class of GSPs havingtwo pairs of collocated joints, whose singularity is treated inthis paper, is one of the more general classes. Their singularityanalysis and geometrical interpretation, is presented here, to thebest of our knowledge, for the first time.