New Bracket Polynomials Associated with the General Gough-Stewart Parallel Robot Singularities

It is well known that the singularities of a Gough- Stewart platform arise when the determinant of the Pl ̈ucker coordinates of the robot leg lines vanish. The direct expansion of this determinant in terms of the configuration of the moving platform leads to an intimidating algebraic expression which is difficult to organize in a manner that facilitates extracting geometric conditions for singularities to occur. The use of Grassmann-Cayley algebra has permitted expressing this de- terminant as a bracket polynomial which is easier to manipulate symbolically. Each monomial in this polynomial is the product of three brackets, 4×4 determinants involving the homogeneous coordinates of four leg attachments. In this paper, we show how to derive, using elementary linear algebra arguments, bracket polynomials where all brackets can be interpreted as reciprocal products between lines. Contrarily to what one might expect, these new bracket polynomials are simpler in general than those previously obtained using Grassmann-Cayley algebra.