Adaptive Search on the Equipotential Surface Using Perturbation Methods for UAV-Aided Multiuser LOS Relaying

This paper studies optimal unmanned aerial vehicle (UAV) placement to establish line-of-sight (LOS) relaying channels for multiple ground nodes in dense urban areas with arbitrary obstacle topology. The key challenge originates from the fact that the terrain obstacles may have arbitrary locations and shapes, and therefore, the placement problems are non-convex with possibly arbitrarily many local optima. Prior works employ over-simplified probabilistic models for the 3D environment, and thus, the LOS condition cannot be guaranteed. Exploiting the properties of the equipotential surface, it is proved that the globally optimal position in 3D lies on the equipotential surface under certain conditions. With this insight, an adaptive search trajectory on the equipotential surface is developed and the closed-form expressions of search directions are derived using perturbation methods. Numerical experiments using real city map data show that the proposed algorithm achieves over 96.7% of the performance of the exhaustive 3D search scheme in all tested scenarios. Additionally, the proposed algorithm only needs a 172-meter search length, while other baseline schemes have dozens of times longer search trajectories.