Singularity-invariant leg rearrangements in Stewart-Gough platforms

Postprint (author’s final draft

Flagged Parallel Manipulators

The conditions for a parallel manipulator to be flagged can be simply expressed in terms of linear dependencies between the coordinates of its leg attachments, both on the base and on the platform. These dependencies permit to describe the manipulator singularities in terms of incidences between two flags (hence, the name “flagged”). Although these linear dependencies might look, at first glance, too restrictive, in this paper the family of flagged manipulators is shown to contain large subfamilies of six-legged and three-legged manipulators. The main interest of flagged parallel manipulators is that their singularity loci admit a well-behaved decomposition, with a unique topology irrespective of the metrics of each particular design. In this paper, this topology is formally derived and all the cells, in the configuration space of the platform, of dimension 6 (non-singular) and dimension 5 (singular), together with their adjacencies, are worked out in detail

Straightening-Free Algorithm for the Singularity Analysis of Stewart-Gough Platforms with Collinear/Coplanar Attachments

Abstract An algorithm to derive the pure condition of any double-planar Stewart-Gough platform in a standard form suitable for comparison is presented. By applying the multilinear properties of brackets directly to the superbracket encoding of the pure condition, no straightening is required. It is then shown that any 3-3 platform has a corresponding 6-6 platform having its same superbracket, meaning that they have identical singularity loci. In general, the superbracket of any doubleplanar platform can be written as a linear combination of the superbrackets of 3-3 platforms, leading to a direct singularity assessment by inspecting the resulting decomposition.

On Plane Cremona Transformations of fixed degree

We study the quasi-projective variety Bird of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir◦d where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bird is connected for each d and Bir◦d is connected when d <7