# Connectivity Maintenance Based on Multiple Relay UAVs Selection Scheme in Cooperative Surveillance
## 3 Multiple Relay UAVs Selection Scheme Based on Fuzzy Optimum Selection
### Receding Horizon Motion Plan Based on UAV Roles  
As the communication network is connected, all UAVs can share surveillance and state information while performing tasks. Considering UAV kinematic constraints, collision avoidance and connectivity preservation in multiple UAVs cooperative surveillance, each UAV can optimize its motion plan by decentralized receding horizon control according to relevant different UAV roles. According to Equation (1), the continuous UAV model can be discretized as follow:  
\[\mathbf{x}_{k+1}=f_{d}(\mathbf{x}_{k},\mathbf{u}_{k})=\mathbf{x}_{k}+T_{S}f(\mathbf{x}_{k},\mathbf{u}_{ k}) \tag{24}\]  
where \(\mathbf{x}_{k}\) is UAV current state vector, and \(\mathbf{u}_{k}\) is control command vector. At each sample instant, each UAV next instant state can be propagated by above equation. In next sample period, each UAV optimize its control sequence based on updated local neighbor information in the same manner.  
The given planning horizon length can be expressed as \(N\), and the control command sequence of UAV \(m\) can be denoted as \(\mathbf{u}_{m,k}=[\mathbf{u}_{m,k}^{0},\mathbf{u}_{m,k}^{1},\dots,\mathbf{u}_{m,k}^{N-1}]\). For relevant UAV roles, the optimization metrics are different. The primary purpose of R UAV is maintaining connectivity by constructing communication links, which is realized by circling over predefined points, and the optimization metric is described as follow:  
\[J^{m}(k)=-\sum_{i=1}^{N}\left(\frac{\|\mathbf{x}_{m,(k+i)}-\mathbf{x}_{m,i}^{c}\|-r_{l }}{r_{l}}\right)^{2} \tag{25}\]  
where \(\mathbf{x}_{m,i}^{c}\) is the assigned circling point, and \(r_{l}\) is the circling radius calculated by UAV kinematic constraints. For AUAV, its primary task is maintaining balance between surveillance payoff and network connectivity, hence AUAV is a particular type of SUAV with additional connection tasks. Correspondingly, the performance of estimated surveillance payoff can be expressed as follow:  
\[J^{m}(k)=\sum_{i=1}^{N}c_{d}^{m}(t_{k+i})=\sum_{i=1}^{N}\sum_{q\in Q_{k+i}^{m} }\xi_{d}^{m}(q,t_{k+i}) \tag{26}\]  
where \(Q_{k+i,N}^{m}\) is the set of grids can be searched by UAV \(m\) in the \(i\)th horizon at time instant \(t_{k}\).  
In order to satisfy the collision avoidance and connectivity preservation constraints, a simple penalty function is designed to penalize the movement that may cause collision or network disruption, and the penalty function can be expressed below:  
\[\small\begin{split}&\small\begin{split}\small\begin{split}\small\begin{split}\small\begin{split}\small\begin{split}\small\begin{split}\small\begin{split}\small\begin{array}{c}\  
penalty function are dynamically adjusted by the role of UAV \(m\), and the optimal control command sequence can be written as:  
\[\mathbf{u}_{m}^{*}=[\mathbf{u}_{m,0}^{*},\mathbf{u}_{m,1}^{*},\ldots,\mathbf{u}_{m,(N-1)}^{*}]= \text{argmax}J_{m}(k) \tag{30}\]  
where \(\mathbf{u}_{m}^{*}\) is current optimization control command sequence, and \(\mathbf{u}_{m,0}^{*}\) is selected as current control command for UAV \(m\).  
Due to the complexity of multiple UAVs cooperative surveillance problem, evolutionary algorithms should be employed to optimize each UAV's motion plan, such as immune algorithm, genetic algorithm, particle swarm optimization, etc. Considering the flexibility, robustness and global optimization capacity, the UAV control command sequence can be solved by PSO with elite mechanism in this work.