# 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
The size of the projected area (\((L_{x},L_{y})\)) can be calculated as  
\[L_{x}=2H\cdot\tan(\frac{fOAx}{2}) \tag{5}\]  
\[L_{y}=2H\cdot\tan(\frac{fOAy}{2}). \tag{6}\]  
Therefore, the spacial resolution (\(sr\)) obtained by taking a picture at height \(H\) is  
\[sr=\frac{l_{x}}{L_{x}}=\frac{l_{x}}{2H\cdot\tan(\frac{fOAx}{2})}, \tag{7}\]  
where the pixel width of the photo is denoted as \(l_{x}\). Similarly, the formula for calculating the vertical spatial resolution (\(R_{y}\)) is given as \(l_{x}/(2\times H\times\tan(fOAy/2))\). Therefore,  
\[H_{\min}\leq H\leq\frac{l_{x}}{2 sr\cdot\tan(\frac{fOAx}{2})} \tag{8}\]where \(H_{\text{min}}\) is the lower bound for the height of UAVs for safety. Hence, the mission requirement imposes a constraint on the maximum height the UAV can fly, which is  
\[H_{\text{max}}=\frac{l_{x}}{2 sr\cdot\tan\left(\frac{fOA}{2}\right)} \tag{9}\]  
Furthermore, we denote the distance between UAV \(m\) and UAV \(m^{\prime}\) in t as \(D_{m,m^{\prime},t}\), which can be expressed as  
\[\mathcal{D}_{m,m^{\prime},t}=\sqrt{\left(X_{m,t}-X_{m^{\prime},t}\right)^{2}+ \left(Y_{m,t}-Y_{m^{\prime},t}\right)^{2}} \tag{10}\]  
We assume that the UAVs should keep a minimal distance of \(\hat{\mathcal{D}}_{\text{min}}\) to avoiding their collision in each \(\mathcal{T}\). Then, we have  
\[\mathcal{D}_{m,m^{\prime},t}\geq\mathcal{D}_{\text{min}},\forall m,m^{\prime} \in\mathcal{M},m\
eq m^{\prime}. \tag{11}\]  
To avoid interference beyond the coverage area, we consider ensuring that the minimum distance between drones is greater than their communication coverage range. Therefore, we introduced communication coverage constraints into path optimization, which can be expressed as:  
\[d_{mm^{\prime}}>=R_{\text{cov}} \tag{12}\]  
where \(d_{mm^{\prime}}\) is the communication distance between UAVs \(m\) and \(m^{\prime}\), and \(R_{c}ov\) is the communication coverage radius. In addition, we can obtain the maximum velocity and maximum acceleration of the UAV. \[(x_{i},y_{i},z_{i})\in\mathcal{B},\quad\forall i=1,2,\ldots,t \tag{4}\]0 \[(x_{i},y_{i},z_{i})\in\mathcal{B},\quad\forall i=1,2,\ldots,t \tag{4}\]1  
During the execution of a mission, constraints on the turning angles of multiple drones need to be considered. Turning a drone requires both time and space, and excessively large or small turning angles can affect the efficiency of the drone's path planning and execution. If the turning angle is too large, the drone requires more time and space to complete the turn, which can reduce its execution efficiency. Conversely, if the turning angle is too small, the drone may deviate from the planned path during execution, resulting in flight errors. \[(x_{i},y_{i},z_{i})\in\mathcal{B},\quad\forall i=1,2,\ldots,t \tag{4}\]2  
where \(\theta_{m}(t)\) represents the heading of the \(m\)-th drone at time \(t\), while \(\Delta\theta\) denotes the maximum turning angle. In addition, to ensure that drones do not leave the coverage area when turning and to control the path-planning trajectory of the drones, avoiding excessively complex path planning, constraints on the turning radius need to be considered in multidrone path planning. \[\left|\left[ \begin{array}{cc}\cos(\theta_{m}(t))&\sin(\theta_{m}(t))\\ -\sin(\theta_{m}(t))&\cos(\theta_{m}(t))\end{array} \right]\right|\geq R,\quad \forall m,t \tag{16}\]  
where \(\Delta x\) and \(\Delta y\) refer to the difference between the \(x\) and \(y\) coordinates of the \(m\)-th drone between time steps \(t\) and \(t-1\), \(\theta_{m}(t)\) is the turn angle of the \(m\)-th drone at time, \(R\) represents the minimum turn radius in drone path planning, and \(\left|\cdot\right|\) is the Euclidean paradigm (L2 paradigm) used to compute the length of a vector obtained by matrix-vector multiplication. For vector \((x,y)\), the Euclidean norm is given by \(\left|(x,y)\right|=\sqrt{x^{2}+y^{2}}\).