On Δ-Transforms

Abstract

Any set of two legs in a Gough–Stewart platform sharing an attachment is defined as a Δ 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 Δ 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 Δ-transforms, and they can be applied sequentially and simultaneously. Although it may seem counterintuitive at first glance, the rearrangement of legs using simultaneous Δ-transforms does not necessarily lead to leg configurations containing a Δ component. As a consequence, the application of Δ-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.