On $\Delta$-Transforms

Abstract

Abstract-Any set of two legs in a Gough-Stewart platform sharing an attachment is defined as a ∆component. This component links a point in the platform (base) to a line in the base (platform). Thus, if the two legs, which are involved in a ∆ component, are rearranged without altering the location of the line and the point in their base and platform local reference frames, the singularity locus of the Gough-Stewart platform remains the same, provided that no architectural singularities are introduced. Such leg rearrangements are defined as ∆-transforms, and they can be applied sequentially and simultaneously. Although it may seem counterintuitive at first glance, the rearrangement of legs using simultaneous ∆-transforms does not necessarily lead to leg configurations containing a ∆component. As a consequence, the application of ∆-transforms reveals itself as a simple, yet powerful, technique for the kinematic analysis of large families of Gough-Stewart platforms. It is also shown that these transforms shed new light on the characterization of architectural singularities and their associated self-motions.

The octahedral manipulator revisited

“…Nevertheless, it is interesting to realize that these offsets can also be introduced simultaneously, not only sequentially. The details of how this operation is performed can be found in [20]. Then, if an offset is simultaneously introduced for the six sets of two legs sharing a double-ball-joint, all joints are split into single spherical joints.…”

“…According to Fig. 6 and the results in [20], the affine relation between leg lengths of the resulting 6-6 platform and the original octahedral manipulator can be expressed as:…”

On quartically-solvable robots

“…Now, observe that Q1 contains a line-line and a pointline component [5]. The multiple spherical joints in the lineline component can be eliminated by using the singularityinvariant transformation presented in [6]. This leads to Q2.…”

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

“…This family of parallel platforms is defined by certain geometric constraints in the location of their leg attachments to the fixed base and/or moving platform such as coincidence and collinearity (see Fig. 1 dinates of the attachments in their local reference frames (see [1] for a non-trivial example).…”