## 2 Materials and Methods
### UAV Localization via Pass Position  
We believe that the UAV swarm has a regular formation at the initial time; that is, each UAV has its SRP. However, various disturbances will occur in actual flight. At this time, if the position of ABP is correct, positioning can be performed by obtaining the relative distance between \(\text{ABP}\ \textbf{p}_{\textbf{n}}=[x_{n},y_{n},z_{n}]^{T}\) and the receiver \(\textbf{p}_{\textbf{c}}=[x_{c},y_{c},z_{c}]^{T}\). For the stability of the transmission link, the signal in the A2A internal link does not carry effective information, so the relative distance cannot be directly transmitted. Therefore, we use passive positioning where each AP passively receives signals and uses AOAI obtained by them using the communication model as the signal feature used in passive positioning. In the condition of knowing the AOAI based on the circular angle theorem: the circular angle subtended by an arc is equal to half of the central angle it subtends, and we can be sure that the receiver is located on the only circle with connection with two ABSs as the chord. Considering that this is in three-dimensional space, the center of it should change from a certain point to a three-dimensional ring. From the vector angle formula, we can obtain  
\[AS(n1,n2)=\frac{(\textbf{p}_{\textbf{c}}-\textbf{p}_{\textbf{n1}})(\textbf{p}_ {\textbf{c}}-\textbf{p}_{\textbf{n2}})}{|(\textbf{p}_  
For convenience of expression, the above formula can be converted into spherical coordinates:  
\[x =r\sin\theta\cos\phi+O_{x}, \tag{17}\] \[y =r\sin\theta\sin\phi+O_{y},\] (18) \[z =r\cos\theta+O_{z} =\frac{1}{3}(rO_{x}sin\theta cos\phi+rO_{y}sin\theta sin\phi), \tag{19}\]  
where \(\theta\) is the angle between the point and positive z-axis and \(\phi\) is the angle between the point and positive x-axis. On the other hand, the radius of circle \(R\) is fixed:  
\[R=r\sin(\frac{\alpha}{2}), \tag{20}\]  
From the above formula, we can easily see that the actual position of AP is above the intersection of the torus, and this is very similar to directly obtaining the geometric model of the distance between ABS and AP, so we use \(\mathbf{O_{1}}=(O_{x},O_{y},O_{z})\) as the virtual air base station (VABS). It is convenient to establish the distance equation from the VABP to the receiver:  
\[\left\{\begin{array}{l}(x_{c}^{2}+y_{c}^{2}+z_{c}^{2}-r_{1}^{2}-R_{1}^{2})^ {2}+4R_{1}^{2}z_{c}^{2}=4r_{1}^{2}R_{1}^{2}\\ (x_{c}^{2}+y_{c}^{2}+z_{c}^{2}-r_{2}^{2}-R_{2}^{2})^{2}+4R_{2}^{2}z_{c}^{2}=4r _{2}^{2}R_{2}^{2}\\ \cdots\\ (x_{c}^{2}+y_{c}^{2}+z_{c}^{2}-r_{k}^{2}-R_{k}^{2})^{2}+4R_{k}^{2}z_{c}^{2}=4r _{k}^{2}R_{k}^{2}\end{array}\right., \tag{21}\]  
where \(r_{k}\) is the inner radius of the moving ring calculated by the K UAV through Equation (16e) and \(R_{k}\) is the outer radius of the moving ring calculated by the K UAV through Equation (20).  
To solve the above nonlinear equation system, after subtracting the nth equation from the first n \(-\) 1 equations, the linear equation is obtained:  
\[AX=b,\] (22a) where \[A=\begin{pmatrix}2f(1,k)&2f(1,k)&2g(1,k)\\ \vdots&\vdots&\vdots\\ 2f(i,k)&2f(i,k)&2g(i,k)\end{pmatrix}, \tag{22b}\] \[f(i,k)=r_{k}^{2}+R_{k}^{2}-r_{i}^{2}-R_{i}^{2},\] (22c) \[g(i,k)=r_{k}^{2}-R_{k}^{2}+R_{i}^{2}-r_{i}^{2},\] (22d) \[b=\begin{pmatrix}(r_{k}^{2}+R_{k}^{2})^{2}-(r_{1}^{2}+R_{1}^{2})^ {2}+r_{1}^{2}-r_{k}^{2}\\ \vdots\\ (r_{k}^{2}+R_{k}^{2})^{2}-(r_{k-1}^{2}+R_{k-1}^{2})^{2}+r_{k-1}^{2}-r_{k}^{2} \end{pmatrix}, \tag{22e}\]  
Solving the above equation by the method of least squares, we obtain  
\[X=(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b, \tag{23}\]  
Therefore, in the case that signal has no valid information, the coordinates of the follower can be obtained if the ABS coordinates are correct.