On the Primal and Dual Forms of the Stewart Platform Pure Condition.

The algebraic characterization of the singularities of a Stewart platform is usually presented as a $\\hbox{6}\\times \\hbox{6}$ determinant, whose rows correspond to the line coordinates of its legs, equated to zero. This expression can be rewritten in a more amenable way, which is known as the pure condition, as sums and products of $\\hbox{4}\\times \\hbox{4}$ determinants, whose rows correspond to the point coordinates of the leg attachments. Researchers usually rely on one of these two expressions to find the geometric conditions associated with the singularities of a particular Stewart platform. Although both are equivalent, it is advantageous to use either line or point coordinates, depending on the platform topology. In this context, an equivalent expression involving only plane coordinates, i.e., a dual expression to that using point coordinates, seems to be missing. This paper is devoted to its derivation and to show how its use is advantageous in many practical cases, mainly because of its surprising simplicity: It only involves the addition of $\\hbox{4}\\times \\hbox{4}$ determinants whose rows are plane coordinates defined by sets of three attachments.