Supplementary material from "Beacon-referenced pursuit for collective motions in three dimensions"

Motivated by real-world applications of unmanned aerial vehicles, this paper introduces a decentralized control mechanism to guide steering control of autonomous agents maneuvering in the vicinity of multiple moving entities (e.g. other autonomous agents) and stationary entities (e.g. fixed beacons or points of references) in a three-dimensional environment. The proposed control law, which can be perceived as a modification of the three-dimensional constant bearing (CB) pursuit law, provides a means to allocate simultaneous attention to multiple entities. We investigate the behaviour of the closed-loop dynamics for a system with one agent referencing two beacons, as well as 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 beacons. 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 three-dimensional setting.