Autonomous Drone Model

This repository contains the simulations based on the recently introduced Autonomous Drone Model [1]. This is an ODE system describing the interactions between $n$ drones. The model is as follows:

$\begin{aligned} \dot{x}_i(t) &= v_i(t), &&i\in \{0, \dots, n-1\}\\ \dot{v}_i(t) &= A_{i}\left(1-\frac{v_{i}(t)}{{V}_{i}}-v_{i}(t)S_i(t)\right)+\frac{H_i(t)}{m_i}, \ && i\in \{0, \dots, n-1\} \\ S_i(t) &={\Huge\{}\begin{aligned} & 0, \\ & \textstyle{\frac{1}{\kappa}\sum_{j=0}^{i-1} K_{j}\exp{\frac{x_i(t)-x_j(t)}{\omega}}}, \end{aligned} && \begin{aligned} &i=0 \\ &i\in \{1, \dots, n-1\} \end{aligned} \end{aligned}$

with initial conditions

$\begin{aligned} &0\leq x_{n-1}(t_0) \leq \ldots \leq x_1(t_0) \leq x_0(t_0),\\ &0 \leq v_i(t_0) \leq V_i,\quad\quad\quad i\in\{0,\ldots,n-1\}. \end{aligned}$

The variable $x_i(t)$ (expressed in meters [m]) describes the position of $i$-th drone (ordered from the drone at the front at time $t$, while variable $v_i$ – its velocity (expressed in $\frac{\text{m}}{\text{s}}$). Parameters $A_i$ and $V_i$ describe the maximum acceleration and the maximum velocity of the $i$-th drone, respectively. Parameters $m_i$ and $K_i$ describe the size of the $i$-th drone: $m_i$ is its mass and $K_i$ is the surface of its cross-section. Parameter $\kappa$ describes the capacity of the air corridor inside the horizon $\omega$, i.e. the distance in front of the drone, in which the preceding drones have a higher impact on the movement, while $H_i$ describes the wind force. According to Battista et al. [2], the wind force acting on a drone can be described by:

$H_i(t)=\pm \frac{1}{2}\rho C_{d}A_{f,i}v_{wind}^{2}(t),$

where $\rho$ is a density of air, $C_{d}$ is a drag coefficient (dimensionless), $A_{f,i}$ is a frontal area of drone $i$ and $v_{wind}$ is a wind velocity. Using formula above, one can compute wind-induced acceleration acting on drone $i$ simply by dividing it by the mass of a drone, in accordance with Newton's second law of motion. The parameters and units are summarized in the table below. It is assumed that all parameters are positive.

[1] A. Lonc, B. Domżał, M.J. Piotrowska (2026). Autonomous Drone Model: a mathematical study. Applied Mathematical Modelling. DOI: 10.1016/j.apm.2026.116750.

[2] A. Battista, D. Ni (2017). Modeling Small Unmanned Aircraft System Traffic Flow Under External Force. Transportation Research Record: Journal of the Transportation Research Board. DOI:10.3141/2626-10.

Citing

If you are using this repository, please cite: this paper.