Self-Assembly of a Class of Infinitesimally Shape-Similar Frameworks

Formation control strategies are fundamentally impacted by the sensing modalities present in the multi-robot team. Infinitesimal shape-similarity describes frameworks for which maintaining the relative angles between robots in formation also maintains the shape up to translation, rotation, and uniform scaling; however, ensuring invariance of the formation to these motions requires that the robots measure a sufficient number of angles, which means that the topology of the frame-work must be carefully designed. In this paper, we investigate the self-assembly of a class of infinitesimally shape-similar frameworks by robots equipped with bearing-only sensors. To accomplish self-assembly, we introduce a rank condition on the shape-similarity matrix for analyzing frameworks; we then use this rank condition to show that triangulations are infinitesimally shape-similar. A graph grammar is presented to assemble triangulations, and a controller is designed to achieve self-assembly of a team of differential-drive robots.