## 4 Problem Statement
### Notations and Formalism  
#### 4.1.1 Environment \(E\)  
An environment \(E=(P,L,W)\) is a three-dimensional finite space composed of the following:  
* A set of points \(P=\{p_{1},...,p_{n}\}\rightarrow\mathbb{R}^{3}\) such as \(\forall i\in[1..n]\); the positions \(p_{i}=(x_{i},y_{i}.z_{i})\in\mathbb{R}^{3}\) of these waypoints are defined by 3D Cartesian coordinates.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Size** & \begin{tabular}{l} **Dimensions** \\ **(m\({}^{2}\))** \\ \end{tabular} & \begin{tabular}{l} **Payload** \\ **(kg)** \\ \end{tabular} & \begin{tabular}{l} **Velocity** \\ **(km/h)** \\ \end{tabular} & \begin{tabular}{l} **Altitude** \\ **(km)** \\ \end{tabular} & **Example (Accessed on 15 March 2023)** \\ \hline Very small & 0.3–0.5 & & \(\leq\)10 & \(<\)0.12 & \begin{tabular}{l} SmartPlane Pro ([[https://www.drohnen.de/tag/tobyrich/](https://www.drohnen.de/tag/tobyrich/)]([https://www.drohnen.de/tag/tobyrich/](https://www.drohnen.de/tag/tobyrich/))) \\ DJI Mavic 3 ([[https://store.dji.com/de/product/dji-mavic-3](https://store.dji.com/de/product/dji-mavic-3)]([https://store.dji.com/de/product/dji-mavic-3](https://store.dji.com/de/product/dji-mavic-3))) \\ \end{tabular} \\ \hline Small & 0.51–2 & \(<\)9 & \(<\)185 & \(<\)0.4 & \begin{tabular}{l} Wingcopter 198 ([[https://wingcopter.com/](https://wingcopter.com/)]([https://wingcopter.com/](https://wingcopter.com/))) \\ Astro ([[https://freeflysystems.com/alta-x](https://freeflysystems.com/alta-x)]([https://freeflysystems.com/alta-x](https://freeflysystems.com/alta-x))) \\ Scorpion ([[https://www.quantum-systems.com/](https://www.quantum-systems.com/)]([https://www.quantum-systems.com/](https://www.quantum-systems.com/))) \\ \end{tabular} \\ \hline Medium & 5–10 & \(<\)200 & \(<\)1.1 & \
\begin{tabular}{l} Trinity F90+ ([[https://www.quantum-systems.com/](https://www.quantum-systems.com/)]([https://www.quantum-systems.com/](https://www.quantum-systems.com/))) \\ Vector ([[https://www.quantum-systems.com/](https://www.quantum-systems.com/)]([https://www.quantum-systems.com/](https://www.quantum-systems.com/))) \\ Alta X ([[https://freeflysystems.com/alta-x](https://freeflysystems.com/alta-x)]([https://freeflysystems.com/alta-x](https://freeflysystems.com/alta-x))) \\ Yangda YD6-1600S ([[https://www.yangdaonline.com/](https://www.yangdaonline.com/)]([https://www.yangdaonline.com/](https://www.yangdaonline.com/))) \\ \end{tabular} \\ \hline Large & \(>\)10 & \(<\)600 & \(<\)5.5 &
\begin{tabular}{l} Primoco UAV One 150 ([[https://uav-stol.com/primoco-uav-one-150/](https://uav-stol.com/primoco-uav-one-150/)]([https://uav-stol.com/primoco-uav-one-150/](https://uav-stol.com/primoco-uav-one-150/))) \\ Feng Ru 3-100 ([[https://ev.buaa.edu.cn/info/1133/3165.htm](https://ev.buaa.edu.cn/info/1133/3165.htm)]([https://ev.buaa.edu.cn/info/1133/3165.htm](https://ev.buaa.edu.cn/info/1133/3165.htm))) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: UAV classification according to the size ([[https://www.e-education.psu.edu/geog892/node/5](https://www.e-education.psu.edu/geog892/node/5)]([https://www.e-education.psu.edu/geog892/node/5](https://www.e-education.psu.edu/geog892/node/5)), accessed on 15 March 2023).  
Figure 5: Workflow of target detection with multi-UAV system.  
* A set of lines \(L=\{(p_{i},p_{j})|p_{i},p_{j}\in P\times P\}\) that connect the waypoints.
* A weighting function \(W:E\rightarrow\mathbb{R}\) associated with the lines of \(E\), such as \(W=\{w(p_{i},p_{j})|p_{i},p_{j}\in P^{2}\text{ and }p_{i}\
eq p_{j}\}\).  
The number of finite elements of \(E\) is the cardinal \(|P|\) and the density (weight) of the environment is \(|L|\). We will denote the shape of the environment by \(s(E)\)