# Connectivity Maintenance Based on Multiple Relay UAVs Selection Scheme in Cooperative Surveillance
## 2 Problem Definition
### Multiple UAVs Cooperative Surveillance Problem Formulation
According to one look detection probability and history detection information, the detection probability of target \(j\) in grid \(q\) can be estimated [13] as:  
\[b_{j}^{m}(\hat{h}^{m}(q,t_{k}))=1-\prod\limits_{n=1}^{N_{U}}\left(1-a_{j}^{n} \right)^{L^{m,n}(q,t_{k}-\tau^{n,m})} \tag{4}\]  
Based on the latest surveillance information, UAV \(m\) plans its motion and decides whether taking one more look to search grid \(q\) based on the estimation detection payoff. If UAV \(m\) decides to take one more look on grid \(q\), the corresponding estimated total detection number on grid \(q\) by UAV \(m\) will be modified as \(\hat{L}^{m}(q,t_{k})=L^{m}(q,t_{k})+1\). Integrating the modified total number \(\hat{L}^{m}(q,t_{k})\) into Equation (4), the estimated detection probability of target \(j\) in grid \(q\) with one more look can be given as:  
\[\hat{b}_{j}^{m}(\hat{h}^{m}(q,t_{k}))=1-(1-a_{j}^{m})^{\hat{L}^{m}(q,t_{k})} \prod\limits_{n=1,n\
eq m}^{N_{U}}\left(1-a_{j}^{n}\right)^{L^{m,n}(q,t_{k}- \tau^{n,m})} \tag{5}\]Equation (4) reflects the estimated detection probability of target \(j\) in grid \(q\) at current instant \(t_{k}\), and Equation (5) represents the estimated detection probability of taking one more look by UAV \(m\) for target \(j\). Subsequently, the estimated improvement of detection probability for target \(j\) if UAV \(m\) performs one more look over grid \(q\) can be calculated as follow:  
\[\hat{\beta}_{j}^{m}(\hat{h}^{m}(q,t_{k}))=\hat{b}_{j}^{m}(\hat{h}^{m}(q,t_{k}) )-b_{j}^{m}(\hat{h}^{m}(q,t_{k})) \tag{6}\]  
Considering the a priori probability of target existence \(p_{j}^{m}(q)\) in grid \(q\), where \(p_{j}^{m}(q)=p_{j}(q)\) is satisfied for each UAV by gathering same initial information, the estimated additional surveillance payoff of UAV \(m\) performing one more look on grid \(q\) can be defined as \(\hat{o}_{d}^{m}(q,t_{k})=\sum\limits_{j\in T}p_{j}(q)\hat{\beta}_{j}^{m}(\hat {h}^{m}(q,t_{k}))\). Hence, the surveillance payoff of UAVs group at \(t_{k}\) time instant in cooperative surveillance tasks can be defined as:  
\[p(t_{k})=\omega\sum\limits_{q\in Q}\sum\limits_{j\in T}p_{j}(q)\cdot(1-\prod \limits_{m=1}^{N_{U}}\left(1-a_{j}^{m}\right)^{L^{m}(q,t_{k})}) \tag{7}\]  
Actually, the surveillance payoff is defined as the product of detection probability and target existence probability. \(\omega\) is a weighting parameter as \(10^{6}\) in this work, and \((1-\prod\
olimits_{m=1}^{N_{U}}\left(1-a_{j}^{m}\right)^{L^{m}(q,t_{k})})\) is the overall detection probability of target \(j\) in grid \(q\) at \(t_{k}\) time instant, which is modified by Equation (4) without communication delay. \(T\) is the set of targets randomly scattering in surveillance region, and \(Q\) is the set of grids detected in surveillance region. In each time instant, neighbor UAVs only need to share total detection number on grid \(q\) to separately update stored history detection information of group. Therefore, when UAV \(m\) decides to take one more look over grid \(q\), the surveillance payoff improvement with single step motion can be estimated. Considering the detection history information of group, the a priori probability of target existence and communication delay, the total surveillance payoff estimation with one more look can be calculated as:  
\[c_{d}^{m}(t_{k})=\omega\sum\limits_{q\in Q^{m}}\hat{c}_{d}^{m}(q,t_{k})=\omega \sum\limits_{q\in Q^{m}}\sum\limits_{j\in T}p_{j}(q)\hat{\beta}_{j}^{m}(\hat {h}^{m}(q,t_{k})) \tag{8}\]  
If just one step motion plan is considered, the optimal control command of UAV \(m\) can be described as:  
\[\mathbf{u}_{m}^{*}(t_{k})=\text{argmax}c_{d}^{m}(t_{k}) \tag{9}\]  
In [27], single UAV's search motion in a certain sequence of steps is optimized by a greedy strategy. Similar to the highest search effectiveness mentioned, the purpose of UAVs motion plan with corresponding constraints is maximizing surveillance payoff, which is optimized by decentralized receding horizon control described in detail in next section.