Beacon-referenced Mutual Pursuit in Three Dimensions

Motivated by real-world applications of unmanned aerial vehicles, this paper introduces a steering control law for a pair of agents maneuvering in the vicinity of a fixed beacon in a 3-dimensional setting. This control law is a modification of the previously studied 3-dimensional constant bearing (CB) pursuit law, in the sense that it incorporates an additional term to allocate attention to the beacon. We investigate the behavior of the closed-loop dynamics for a two-agent mutual pursuit system wherein each agent employs the beacon-referenced CB pursuit law with regards to the other agent and a stationary beacon. Under certain assumptions on the associated control parameters, we demonstrate that this problem admits circling equilibria with agents moving on circular orbits with a common radius, in planes perpendicular to a common axis passing through the beacon. As the common radius and distances from the beacon are determined by the choice of parameters in the pursuit law, this approach provides a means to engineer desired formations in a 3-dimensional setting.