Singularity Analysis of Rigid Directed Bearing Graphs for Quadrotor Formations

The decentralization of formations using onboard sensing is important for multirobot systems, improving the robustness and independence of fleet operations. Bearing measurements (obtainable from embedded cameras) are an attractive choice for use in decentralized formation control, however, this requires that the formation framework be bearing rigid. Rigidity may be checked numerically for a given formation framework, however, it remains difficult to determine the geometric conditions under which otherwise rigid formations become flexible. This article models the sensor and robot constraints in bearing formations of quadrotors as a kinematic mechanism with analogous properties to find geometric conditions for the degeneration of bearing rigidity (singularities) and the resulting uncontrollable motions. A classification of singularities based on graph substructures is developed, and it is shown that arbitrarily large formations may be designed for which all singularities lie within a known set of geometric conditions. An application on how to use the knowledge of all singularity cases in a formation for singularity-free control maintenance is provided.