Sign-elevation angle rigidity and its application to bearing-only formation control in agents' local frames

Abstract

In this paper, we propose sign-elevation angle rigidity and design bearing-only control laws in agents' local coordinate frames without any orientation synchronization or estimation algorithms. Moreover, we also establish almost global stability results for formation control, with elevation angle measurements that can be obtained using bearing-only sensors in the agents' local frames.  Elevation angle rigidity-based control laws can only guarantee local stability results, and existing results do not address flip, flex, and reflection ambiguities that can arise in elevation angle rigid configurations. Hence, we first establish the theory of sign-elevation angle rigidity to uniquely characterize formation shapes, up to translation and rotation, by employing elevation angle and signed-area constraints for planar frameworks, while for 3-D frameworks, elevation angle constraints along with signed-volume constraints are used. Then, we propose control laws for single integrators and prove almost global stabilization to the desired formation shapes using bearing-only measurements when the desired formation is obtained by triangulated sub-formations for 2-D graphs and tetrahedral sub-formations for 3-D graphs. Control laws for non-holonomic agents are also analyzed. Finally, simulations are provided to validate the obtained results.