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. The Segre embedding arises from a representation of the special Euclidean group in three dimensions which has degree 40. We also consider the special Euclidean group in other dimensions.