# Connectivity Maintenance Based on Multiple Relay UAVs Selection Scheme in Cooperative Surveillance
## 2 Problem Definition
### Multiple UAVs Cooperative Surveillance Problem Formulation  
In this work, a remote and large area surveillance mission is designed, and there are some important and unknown targets scattering in this region. Real-time surveillance information can be  
Figure 3: Schematic of coverage grids with UAV motions. Figure 2: Sensor model in multiple UAVs cooperative surveillance. gathered and sent back to ground station via multihop communications, which widely exists in disaster rescue, area monitoring and situation awareness. Combined with history detection information and surveillance payoff estimation, a multiple UAVs cooperative surveillance framework is constructed in this section. Considering that there are \(N_{U}\) homogeneous UAVs performing cooperative surveillance tasks, and the set of UAVs is defined as \(U=\left\{U_{1},U_{2},\ldots,U_{N_{U}}\right\}\). There are \(N_{T}\) targets randomly scattering in surveillance region. Similarly, the set of targets is defined as \(T=\left\{T_{1},T_{2},\ldots,T_{N_{T}}\right\}\), and the target existence probability is normally distributed without loss of generality. Surveillance region is assumed to be a \(L_{X}\times L_{Y}\) rectangle region on a ground plane, which is uniformly divided into \(N_{X}\times N_{Y}\) grids of the same size, and it is assumed that each grid is identified with its center for simplicity. In remote surveillance scenario, since some UAVs are gradually away from ground station for searching and detecting unknown regions, it is essential to establish UAV-to-UAV communication links for information transmission. Thus, some UAVs need to change roles and serve as relays to guarantee communication connection, and a dynamic communication links chain is constructed to balance surveillance payoff and network connectivity, which is specifically described in next section. According to [13], the probability that UAV \(m\) detects target \(j\) with one look in a grid where the target \(j\) exists is defined as \(a_{j}^{m}\), and \(l^{m}(q,t_{k})\) is the number of looks on grid \(q\) performed by UAV \(m\) at \(t_{k}\) time instant. Hence, the history detection numbers on grid \(q\) performed by UAV \(m\) can be recorded as:  
\[h^{m}(q,t_{k})=\left\{l^{m}(q,t_{1}),l^{m}(q,t_{2}),\ldots,l^{m}(q,t_{k})\right\} \tag{3}\]  
Consequently, \(L^{m}(q,t_{k})=\sum\limits_{i=1}^{k}l^{m}(q,t_{i})\) is the corresponding total number of looks on grid \(q\). As communication network is connected, and all detection history information over grids can be shared between UAVs via single or multiple hops communications. It is assumed that each UAV independently looks over the same grid at different surveillance moments, and the detection process for the same grid by different UAVs is independent. Thus, history detection information reflects the entire surveillance process, and the uncertainty of target existence in the grid gradually reduces with more looks. As UAVs share surveillance information via multiple hops communications, UAVs receive the latest detection history information from its neighbors and perform relative surveillance motion. Considering the communication delay, UAV \(m\) receives the detection history number that UAV \(n\) has searched on grid \(q\) can be expressed as \(h^{m,n}(q,t_{k}-\tau^{n,m})\) at \(t_{k}\) time instant, where \(\tau^{n,m}\) is the arbitrary and finite communication delay, and \(L^{m,n}(q,t_{k}-\tau^{n,m})\) is the received corresponding total number of looks on grid \(q\) of UAV \(n\). Hence, detection history information of the group stored by UAV \(m\) at \(t_{k}\) time instant can be recorded as \(\hat{h}^{m}(q,t_{k})=\left\{h^{m,n}(q,t_{k}-\tau^{n,m}),n\in U\right\}\).