Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations

TL;DR

This paper develops methods to compute optimal UAV trajectories for serving traffic, using Hamilton-Jacobi equations and quadratic traffic models, with practical algorithms for real data application.

Contribution

It introduces closed-form solutions for quadratic traffic models and gradient-based algorithms for bi-phase traffic, enabling efficient trajectory optimization.

Findings

Closed-form formulas for quadratic traffic scenarios.

Low-complexity algorithms for bi-phase traffic models.

Effective trajectory simulation from real traffic data.

Abstract

We consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data.