# Maximizing UAV Coverage in Maritime Wireless Networks: A Multiagent Reinforcement Learning Approach
## 3 System Model and Problem Statement
### System Model
#### 3.1.2 Energy Model  
During mission execution, each drone has a different battery-life limit. Failure to consider battery-life limits may result in certain drones losing control due to battery depletion during task execution and thus being unable to complete the task, affecting coverage and task performance. Therefore, to ensure that each drone can complete the task and return, constraints on battery-life limits need to be considered.  
Taking into account factors such as flight energy consumption and turning energy consumption, we define the energy constraint of each drone as its total energy consumption throughout the flight mission, which cannot exceed its energy capacity. Let the energy capacity of each drone be denoted as \(C\), the energy consumption rate be denoted as \(\mathbf{r}\), the total time denoted as \(T\), and the initial energy level denoted as \(E_{0}\). The energy level (\(E_{t}\)) of a drone at time \(t\) can be calculated using the following equation:  
\[E_{t}=E_{0}-\int_{0}^{t}\mathbf{r}\bigg{\|}\frac{dx}{dt}\bigg{\|}dt \tag{17}\]  
where \(x\) is the position vector of the drone, \(\big{|}\frac{dx}{dt}\big{|}\) is the magnitude of the drone's velocity, and \(t\) is the current time.  
Therefore, for the power constraint of each UAV can be described using the following formula:  
\[\int_{0}^{T}\mathbf{r}\bigg{\|}\frac{dx}{dt}\bigg{\|}dt\leq C-E_{0} \tag{18}\]  
Considering the flight energy consumption, turning energy consumption, and communication energy consumption, let the speed of UAV \(i\) be \(V_{i}\) and its flight energy consumption at moment \(t\) be \(E_{i}(t)\); then:  
\[\int_{0}^{t_{f}}E_{m}(t)dt\leq E_{m,max} \tag{19}\]  
\[\int_{0}^{t_{f}}E_{m}^{turn}(t)dt\leq E_{m,max}^{turn} \tag{20}\]  
\[\int_{0}^{t_{f}}E_{mm^{\prime}}^{comm}(t)dt\leq E_{mm^{\prime},max}^{comm} \tag{21}\]  
where \(t_{f}\) denotes the moment when the UAV ends its flight, \(E_{m,max}\) is the maximum energy consumption of UAV \(m\), \(E_{m,max}^{turn}\) is the maximum energy consumption of UAV \(m\) for a turn, and \(E_{mm^{\prime},max}^{comm}\) is the maximum communication energy consumption between UAV \(m\) and UAV \(m^{\prime}\).  
\[E_{m,max}^{turn}+E_{mm^{\prime},max}^{comm}+E_{m,max}\leq\int_{0}^{T}\mathbf{ r}\bigg{\|}\frac{dx}{dt}\bigg{\|}dt \tag{22}\]  
Therefore, our goal is to find a set of paths \(\left((x_{m,1},y_{m,1},z_{m,1}),(x_{m,2},y_{m,2},z_{m,2}),\ldots,\right.\)\(\left.(x_{m,t},y_{m,t},z_{m,t})\right)\) that maximizes the covered area. Specifically, our objective is to maximize the following expression, considering the energy consumption and turn constraints:  
\[\frac{1}{N}\sum_{i=1}^{N}\left((x_{i},y_{i},z_{i})\in\mathbf{A}\right) \tag{23}\]  
where \(N\) represents the total number of grid cells in the grid environment. The formula for maximizing the coverage rate describes how to evaluate the quality of a set of paths. \(\mathbf{A}\) denotes the set of all non-obstacle and non-restricted areas, where \((x_{i}\in A)\) is an indicator function that determines whether a path \((x_{i})\) can cover a non-obstacle and non-restricted area. \(\sum_{i=1}^{N}\left(x_{i}\in\mathbf{A}\right)\) indicates the number of cells that the set of paths \((x_{1},x_{2},\ldots,x_{m})\) can cover.