Numerical modeling of controlled motion of a quadcopter drone

Abstract

The paper focuses on the use of control theory tools to develop a mathematical model of the controlled movement of a quadcopter drone. The purpose of the work is to perform a numerical analysis of the proposed mathematical model of a quadcopter based on an integrated systems approach, which consists in the sequential implementation of several stages of research. These stages are the study of the subject area; formalization of the model; formulation of the optimization problem and clarification of its properties; construction, based on the apparatus of functional analysis and optimization theory, of computational procedures and software for solving the problem of optimizing the drone-quadcopter's flight trajectory; development of techniques and practical recommendations for creating innovations. The research yields several significant results. Firstly, a mathematical model for the controlled motion of the drone-quadcopter is proposed. Additionally, an indirect scheme is presented to approximate the optimal flight trajectory problem, incorporating necessary optimality conditions based on Pontryagin's maximum principle and the penalty function method, which accounts for phase constraints on the drone's degrees of freedom. Computational algorithms are developed and implemented, including an adapted fourth-order Runge-Kutta method for solving auxiliary Cauchy problems and the conditional gradient method for optimizing the solution using the penalty function approach. The corresponding software implementation enables efficient computation of numerical solutions with the desired accuracy, along with practical convergence and effectiveness analysis of the developed algorithms. The research results can be applied in analyzing and studying the behavior of other types of unmanned aerial vehicles. The research was partially funded by the Ministry of Foreign Affairs of the Czech Republic under the project № 23-PKVV-UM-4 “Support for improving the quality of teaching, research and international activities at Oles Honchar Dnipro National University (DNU)” implemented by Charles University and DNU.