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

Abstract

Abstract-The algebraic characterization of the singularities of a Stewart platform is usually presented as a 6 × 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, known as the pure condition, as sums and products of 4 × 4 determinants whose rows correspond to the point coordinates of the legs 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, 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 4 × 4 determinants whose rows are plane coordinates defined by sets of three attachments.

The octahedral manipulator revisited

“…4, when the supporting planes of the triangles P 1 P 2 P 4 , P 2 P 3 P 5 , P 3 P 1 P 6 , and P 4 P 5 P 6 intersect in a single point, the manipulator is in a singular pose [19]. Alternatively, this condition can be expressed as [21]: Now, as in the previous section, let us suppose that the location of the joints are modified so that the lengths of the legs in their new locations are related to those of the original legs through the relation (4). Differentiating (4) with respect to time and substituting (5) in the result, we get…”

Simplified Voronoi Diagrams for Motion Planning of Quadratically-Solvable Gough–Stewart Platforms

“…The pure condition of this kind of platforms factorizes into 4 × 4 determinants which involve the homogeneous coordinates of sets of four attachments [3]. When the coordinates of the leg attachments in the moving platform are expressed in terms of the position and orientation of the platform, x and Θ respectively, three kinds of determinants arise: -Plane-Point determinants: They involve three attachments of the fixed base and one of the moving platform.…”