## 2 Materials and Methods  
### UAV Model  
A collective task is considered to be carried out by \(N\) homogeneous UAVs flying in three-dimensional space. Each UAV has the same hardware installment and software configuration, as well as owns the same communication capability. The communication capability of each UAV can be measured by its communication radius \(D\) (\(D\) is a positive constant). According to the existing literatures on UAVs, the dynamics of each UAV is described by the following differential equation [26, 27],  
\[ \begin{array}{l}\dot{x}_{i}(t)=\theta_{i}(t)\cos\gamma_{i}(t)\cos\psi_{i}(t) \\ \dot{y}_{i}(t)=\dot{\theta}_{i}(t)\cos\gamma_{i}(t)\sin\psi_{i}(t)\\ \dot{h}_{i}(t)=\dot{\theta}_{i}(t)\sin\gamma_{i}(t)\\ \dot{\sigma}_{i}(t)=g\cdot(n_{ii}(t)-\sin\gamma_{i}(t))\\ \dot{\psi}_{i}(t)=g\cdot n_{hi}(t)/(\dot{\theta}_{i}(t)\cos\gamma_{i}(t))\\ \dot{\gamma}_{i}(t)=g\cdot(n_{vi}(t)-\cos\gamma_{i}(t))/\dot{\theta}_{i}(t), \hskip 28.452756 pti=1,\cdots,N\end{array}  \tag{1}\]  
where \(i\) denotes index for the \(i\)th UAV, \(x_{i}(t)\) is the \(i\)th UAV's downrange, \(y_{i}(t)\) is the \(i\)th UAV's crossrange, \(h_{i}(t)\) is the \(i\)th UAV's altitude, \(\dot{\theta}_{i}(t)>0\) is the \(i\)th UAV's speed, \(\dot{\psi}_{i}(t)\in(-\frac{\pi}{2},\frac{\pi}{2})\) is the \(i\)th UAV's heading angle, \(\gamma_{i}(t)\in(-\frac{\pi}{2},\frac{\pi}{2})\) is the \(i\)th UAV's flight path angle, \(n_{hi}(t)\) is the \(i\)th UAV's horizontal load factor, \(n_{vi}(t)\) is the \(i\)th UAV's vertical load factor, \(n_{ti}(t)\) is the \(i\)th UAV's tangential load factor, and \(g\) is the constant acceleration of gravity. For brevity, the proposed simplified model (1) of UAV \(i\) is shown in Figure 1. Generally speaking, the behavior of UAV \(i\) is steered under four forces: lift force, drag force, weight, and thrust force. \(T_{i}(t)\) is the magnitude of the thrust, \(L_{i}(t)\) is the magnitude of the lift force, \(D_{i}(t)\) is the magnitude of the drag force, and \(W_{i}(t)\) is the magnitude of the gravity. Heading angle \(\psi_{i}(t)\) describes whether the aircraft is turning left or right. Flight path angle \(\gamma_{i}(t)\) describes whether the aircraft is climbing or descending. Bank angle \(\mu_{i}(t)=\arctan\frac{\eta_{hi}(t)}{n_{vi}(t)}\) describes whether the aircraft is rolling clockwise or counterclockwise. The three-dimensional aircraft model not only considers the longitudinal plane, but also the lateral plane, thus the control variable \([L_{i}(t),D_{i}(t),T_{i}(t)]\) is replaced with \([n_{ii}(t),n_{hi}(t),n_{vi}(t)]\) in this model (1). Then, the state of UAV \(i\) at time \(t\) can be expressed by its position vector \(p_{i}(t)=[x_{i}(t),y_{i}(t),h_{i}(t)]^{T}\) and its velocity vector  
\[q_{i}(t)=\left[\begin{array}[  
Let the control input of UAV \(i\) at time \(t\) is denoted as \(u_{i}(t)=[n_{ti}(t),n_{hi}(t),n_{vi}(t)]^{T}\). Then, Equation (4) can be rewritten as  
\[\dot{q}_{i}=g\left[ \begin{array}{ccc}\cos\gamma_{i}\cos\psi_{i}&-\sin\psi_{i}& -\sin\gamma_{i}\cos\psi_{i}\\ \cos\gamma_{i}\sin\psi_{i}&\cos\psi_{i}&-\sin\gamma_{i}\sin\psi_{i}\\ \sin\gamma_{i}&0&\cos\gamma_{i}\end{array} \right]\left[ \begin{array}{c}n_{ ti}\\ n_{hi}\\ n_{vi}\end{array} \right]-G, \tag{5}\]  
where \(G=[0,0,g]^{T}\). Then, the simplified point-mass model of UAV \(i\) can be rewritten by  
\[\dot{p}_{i}(t)=q_{i}(t) \tag{6}\] \[\dot{q}_{i}(t)=gH_{i}^{T}(t)u_{i}(t)-G,\hskip 28.452756 pti=1, \cdots,N\]  
where \(H_{i}(t)=[t_{i}(t),n_{i}(t),h_{i}(t)]^{T}\). The definitions of \(t_{i}(t)\), \(n_{i}(t)\), and \(h_{i}(t)\) are, respectively,  
\[t_{i}(t)=\left[ \begin{array}{c}\cos\gamma_{i}(t)\cos\psi_{i}(t)\\ \cos\gamma_{i}(t)\sin\psi_{i}(t)\\ \sin\gamma_{i}(t)\end{array} \right], \tag{7}\]  
\[n_{i}(t)=\left[ \begin{array}{c}-\sin\psi_{i}(t)\\ \cos\psi_{i}(t)\\ 0\end{array} \right], \tag{8}\]  
and  
\[h_{i}(t)=\left[ \begin{array}{c}-\sin\gamma_{i}(t)\cos\psi_{i}(t)\\ -\sin\gamma_{i}(t)\sin\psi_{i}(t)\\ \cos\gamma_{i}(t)\end{array} \right]. \tag{9}\]  
Obviously, \(H_{i}(t)\) is an orthogonal matrix.