Exceptional Stewart--Gough Platforms, Segre Embeddings, and the Special Euclidean Group

Stewart--Gough platforms are mechanisms which consist of two rigid objects, a base and a platform, connected by six legs via spherical joints. For fixed leg lengths, a generic Stewart--Gough platform is rigid with 40 assembly configurations (over the complex numbers), while exceptional Stewart--Gough platforms have infinitely many assembly configurations and thus have self-motion. We define a family of exceptional Stewart--Gough platforms called Segre-dependent Stewart--Gough platforms which arise from a linear dependency of point-pairs under the Segre embedding and compute an irreducible decomposition of this family. We also consider Stewart--Gough platforms which move with two degrees of freedom. Since the Segre embedding arises from a representation of the special Euclidean group in three dimensions which has degree 40, we consider the special Euclidean group in other dimensions and compute spatial Stewart--Gough platforms that move in 4-dimensional space.