# Optimal Trajectories of a UAV Base Station Using Lagrangian Mechanics

## TL;DR

This paper formulates the optimal UAV base station trajectory problem using Lagrangian mechanics, deriving closed-form solutions for quadratic traffic and proposing an iterative algorithm for more complex traffic patterns, with numerical validation.

## Contribution

It introduces a novel application of Lagrangian mechanics to optimize UAV trajectories and develops an algorithm for bi-phase traffic scenarios, advancing UAV mobility optimization methods.

## Key findings

- Closed-form solutions for quadratic traffic intensity.
- An iterative algorithm for bi-phase traffic optimization.
- Numerical results demonstrating improved trajectories over MPC.

## Abstract

In this paper, we consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV) Base Station (BS). We consider a map characterized by a traffic intensity of users to be served. The UAV BS must travel from a given initial location at an initial time 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. When the traffic intensity is quadratic (single-phase), we derive closed-form formulas for the optimal trajectory. When the traffic intensity is bi-phase, we provide necessary conditions of optimality and propose an Alternating Optimization Algorithm that returns a trajectory satisfying these conditions. The Algorithm is initialized with a Model Predictive Control (MPC) online algorithm. Numerical results show how we improve the trajectory with respect to the MPC solution.