# A Coalition Formation Game Approach for Efficient Cooperative Multi-UAV Deployment
## 2 System Model and Problem Formulation
Therefore, motivated by the work in [25], we adopt expressions to estimate the energy consumption for transmitting and receiving of each sensor node through RF signals, which are given by:  
\[ \begin{split} Receiving:& E_{r}=E_{0},\\ Transmitting:& E_{t}(d_{i,j})=\begin{cases}E_{0}+ \varepsilon_{s}\cdot d_{i,j}^{2},&d_{i,j}<d_{0},\\ E_{0}+\varepsilon_{t}\cdot d_{i,j}^{4},&d_{i,j}\geq d_{0},\end{cases}\end{split}  \tag{3}\]  
where \(d_{i,j}\) represents the distance between UAV \(i\) and \(j\), \(E_{0}\) is the energy consumed by radio when performing transmitting and receiving. \(\varepsilon_{s}\) and \(\varepsilon_{t}\) represents the energy consumed by transmitting one bit of data at a shorter distance and at a longer distance, respectively. Thus, consider the data transmission through link of UAV \(i\) and UAV \(j\), the overall energy consumption per bit of data is derived as follows:  
\[E(d_{i,j})=E_{r}+E_{t}(d_{i,j}). \tag{4}\]  
Figure 1: A diagram of unmanned aerial vehicle (UAV) coverage in a mission area. In a coalition, cluster head UAV plays an important role in connecting coalition members and the upper central UAV, so as to achieve data transmission. Suppose that the overall energy consumption function of the whole network is defined as follows:  
\[E_{0}=\sum_{m=1}^{M}\Big{(}\sum_{n\in\text{CO}_{m}}\left(E(d_{n,cl_{m}})\cdot U_{ \mathcal{N}}\right)+E(d_{0,cl_{m}})\cdot U_{CO_{m}}\Big{)}, \tag{5}\]  
where \(cl_{m}\) represents the coalition \(m\)'s cluster head UAV, \(U_{n}\) and \(U_{CO_{m}}\) is UAV \(n\) and UAV set \(CO_{m}\)'s mission reward, indicating the data traffic they collect. Equation (5) contains two items: the first item represents the energy consumption of data transmission, which is collected from coalition \(m\)'s cluster member UAV to \(m\)'s cluster head UAV. The second item represents the energy consumption of data transmission, which is transmitted from coalition \(m\)'s cluster head UAV to the upper central UAV. Therefore, Equation (5) reflects the overall energy consumption of the whole UAV network when all data are transmitted to the central UAV. From the above formulation, it can be concluded that UAV deployment is affected by the importance of ground data traffic distribution, while the energy consumption of the path is also limited by the communication distance between UAVs. When the battery can't support long distance UAV communication, the deployment of UAV coverage can be severely affected, making our proposed problem fall into a dilemma. It is meaningful to accurately characterize the relationships between coverage reward and energy consumption. Therefore, according to Equations (1) and (5), we have the whole UAV network's coverage utility:  
\[U1=U_{\mathcal{N}}-\rho\cdot E_{0}, \tag{6}\]  
where \(\rho\) represents the weight coefficient, indicating the importance of reward and transmission energy consumption. Here, \(\rho\) is calculated and measured from parameters of UAVs and mission requirements environment. Therefore, the setting of \(\rho\) decides the mission requirements and has a strong reference value of weighing the gain and cost. It should be pointed out that some constraints for both variables can be set under the condition of driven mission, such as a guarantee of minimal reward and limitation of maximal energy consumption. For intuitive comparison, the object of this model is maximizing the whole UAV network's coverage utility through adjusting UAVs' state:  
\[\mathcal{G}:s_{n}^{opt}=\arg\max U1,n\in\mathcal{N}. \tag{7}\]  
Here, solving \(\mathcal{G}\) can obtain an efficient multi-UAV deployment and coalition transmission mechanism, which makes UAVs achieve better coverage reward without producing much energy consumed by data transmitting and receiving.