# Maximizing UAV Coverage in Maritime Wireless Networks: A Multiagent Reinforcement Learning Approach
## 3 System Model and Problem Statement
### System Model  
#### 3.1.1 UAV Model  
Let \(\mathcal{M}=\{1,2,3,\ldots,M\}\) denote the set of UAVs. At each time step (\(t\)), the flight direction of UAV \(m\in\mathcal{M}\) is determined by the turn angle (\(\theta_{m,t}\in[0,2\pi)\)). The flight distance is determined by \(d_{m,t}\in[0,d_{\max}]\), where \(d_{\max}\) is the maximum allowed flight distance, and UAVs cannot exceed the boundaries of the target area. The trajectory of each drone can be represented as:  
\[\mathbf{P}_{m}=(x_{m,1},y_{m,1},z_{m,1}),(x_{m,2},y_{m,2},z_{m,2}),\ldots,(x_{ m,t},y_{m,t},z_{m,t}) \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
3.1 System Model & \\ \hline \(m_{r}\mathcal{M}\) & Index, set of UAVs \\ \(t_{r}\mathcal{T}\) & Time slot \\ \((x,y,z),(X,Y,Z)\) & Drone coordinates \\ \(\mathbf{P}\) & UAV path collection \\ \(\mathbf{F}\) & Indicator for UAV trajectory avoiding no-fly zones \\ \(\mathbf{O}\) & Indicator for UAV trajectory avoiding obstacles \\ \(\theta\) & Flight steering angle \\ \((L_{x},L_{y})\) & Shadow area \\ \(\mathcal{B}\) & Specified airspace boundary \\ \(H\) & Flying height \\ \(sr\) & Spatial resolution \\ \(\mathbf{R}\) & Turning radius \\ \(\mathcal{D}\),\(\delta\) & Distance between UAVs \\ \(V\),\(\mathcal{A}\) & Speed and acceleration \\ \(E\) & Energy consumption of UAVs \\ \(C\) & Battery capacity \\ \(\mathbf{r}\) & Battery consumption rate \\ \(T\) & Total time \\ \(\mathcal{A}\) & Set of non-obstacle and non-restricted areas \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of symbols used in this chapter. where \(\mathbf{P}_{m}\) is the trajectory of UAV \(m\), and \((x_{m,t},y_{m,t},z_{m,t})\) represents the position of UAV \(m\) at time step \(t\). When performing coverage tasks, UAVs are not allowed to enter no-fly zones and pass through obstacles. Let \(F=\{f_{1},f_{2},\ldots,f_{n}\}\) represent the set of no-fly zones and \(O=\{o_{1},o_{2},\ldots,o_{k}\}\) represent the set of obstacles. The UAV trajectories should avoid crossing \(F\) and \(O\). The constraints can be formulated as follows: The UAV trajectory should not intersect any no-fly zone:  
\[ \begin{array}{l}(x_{m,i},y_{m,i},z_{m,i}),(x_{m,i+1},y_{m,i+1},z_{m,i+1}) \
otin\mathbf{F},\\ \forall m\in\mathcal{M},\forall i=1,2,\ldots,t-1\end{array}  \tag{2}\]  
The UAV trajectory should not intersect any obstacle:  
\[ \begin{array}{l}(x_{m,i},y_{m,i},z_{m,i}),(x_{m,i+1},y_{m,i+1},z_{m,i+1}) \
otin\mathbf{O},\\ \forall m\in\mathcal{M},\forall i=1,2,\ldots,t-1\end{array}  \tag{3}\]  
where \(t\) is the number of time steps taken by UAV \(m\) in its trajectory. The notation \((x_{m,i},y_{m,i},z_{m,i})\) represents the position of UAV \(m\) at time step \(i\) in the 3D space. The constraints ensure that each UAV's trajectory avoids both the no-fly zones and obstacles during the coverage tasks. Subsequently, the trajectory of the UAV must not exceed the specified airspace boundary, which is denoted as  
\[\mathcal{B}=\{(x_{m,1},y_{m,1},z_{m,1}),(x_{m,2},y_{m,2},z_{m,2}),\ldots,(x_{m,b},y_{m,b},z_{m,b})\}.\]  
Each point (\(r_{i}\)) in \(\mathcal{B}\) represents a 3D coordinate in the airspace. To ensure that the UAV remains within the defined airspace at all times, the position of the UAV at each time step (\(i\)) should satisfy the following constraint:  
\[(x_{i},y_{i},z_{i})\in\mathcal{B},\quad\forall i=1,2,\ldots,t \tag{4}\]  
This constraint ensures that at each time step, the position of the UAV is within the specified airspace boundary defined by the \(\mathcal{B}\) set. It guarantees that the UAV remains within the specified airspace during its entire trajectory. A UAV flying at a given height (\(H\)) from the ground acquires an image corresponding to a specific portion of the area (projected area). The size of the projected area depends on the height (\(H\)) and the angle of view (\(fOA\)).