# Resilient Formation Reconfiguration for Leader-Follower Multi-UAVs
## 3 Problem Formulation  
Consider the following state equation for the \(i\)th follower of the leader-follower multi-UAVs:  
\[ \begin{bmatrix}\dot{V}_{i}\\ \dot{\chi}_{i}\\ \dot{\phi}_{i}\\ \dot{x}_{i}\\ \dot{y}_{i}\\ \dot{z}_{i}\end{bmatrix} = \begin{bmatrix}-g\sin\chi_{i}\\ \frac{-g\cos\chi_{i}}{V_{i}}\\ 0\\ V_{i}\cos\chi_{i}\cos\phi_{i}\\ V_{i}\cos\chi_{i}\sin\phi_{i}\\ V_{i}\sin\chi_{i}\end{bmatrix} + \begin{bmatrix}1&0&0\\ 0&\frac{1}{V_{i}}&0\\ 0&0&\frac{1}{V_{i}\cos\chi_{i}}\\ 0&0&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}a_{1i}\\ a_{2i}\\ a_{3i}\end{bmatrix} ,i=1,\ldots,N. \tag{1}\]  
Therein, \(N\) is the amount of the followers. \(V_{i}\), \(\chi_{i}\), \(\phi_{i}\) stand for the linear velocity, the pitch angle and the yaw angle of the follower. \(q_{i}= \begin{bmatrix}x_{i}&y_{i}&z_{i}\end{bmatrix} ^{T}\) stands for the 3D coordinate of the follower. \(a_{i}= \begin{bmatrix}a_{1i}&a_{2i}&a_{3i}\end{bmatrix} ^{T}\) is the control input consisting of the horizontal acceleration, the vertical acceleration and the angular acceleration of the follower as [34]. In practice, the horizontal acceleration \(a_{1i}\) and the vertical acceleration \(a_{2i}\) can be obtained by decomposing the available linear acceleration while the angular acceleration \(a_{3i}\) can be obtained directly. \(g\) stands for the gravitational acceleration constant. Denote \(p_{i}= \begin{bmatrix}\dot{x}_{i}&\dot{y}_{i}&\dot{z}_{i}\end{bmatrix} ^{T}\). From (1), one can obtain  
\[ \begin{split}\ddot{q}_{i}&=\dot{p}_{i}\\ &=\begin{bmatrix}\dot{V}_{i}\cos\chi_{i}\cos\phi_{i}-V_{i}\sin\chi_{i} \dot{\chi}_{i}\cos\phi_{i}-V_{i}\cos\chi_{i}\sin\phi_{i}\dot{\phi}_{i}\\ \dot{V}_{i}\cos\chi_{i}\sin\phi_{i}-V_{i}\sin\chi_{i}\dot{\chi}_{i}\sin\phi_{i} +V_{i}\cos\chi_{i}\cos\phi_{i}\dot{\phi}_{i}\\ \dot{V}_{i}\sin\chi_{i}+V_{i}\dot{\chi}_{i}\dot{\chi}_{i}\end{bmatrix}.\end{split}  \tag{2}\]  
Substitute \(\dot{V}_{i}=-g\sin\chi_{i}+a_{1i}\), \(\dot{\chi}_{i}=\frac{-g\cos\chi_{i}}{V_{i}}+\frac{1}{V_{i}}a_{2i}\), \(\dot{\phi}_{i}=\frac{1}{V_{i}\cos\chi_{i}}a_{3i}\), and one can obtain  
\[ \begin{split}\ddot{q}_{i}&=\dot{p}_{i}\\ &=\begin{bmatrix}0\\ 0\\ -g\end{bmatrix}+\begin{bmatrix}\cos\chi_{i}\cos\phi_{i}&-\sin\chi_{i}\cos\phi_{i}&- \sin\phi_{i}\\ \cos\chi_{i}\sin\phi_{i}&-\sin\chi_{i}\sin\phi_{i}&\cos\phi_{i}\\ \sin\chi_{i}&\cos\chi_{i}&0\end{bmatrix}\begin{bmatrix}a_{1i}\\ a_{2i}\\ a_{3i}\end{bmatrix}.\end{split}  \tag{3}\]  
Denote \(a_{i}=A_{i}^{-1}(u_{i}-b)\), where \(A_{i}= \begin{bmatrix}\cos\chi_{i}\cos\phi_{i}&-\sin\chi_{i}\cos\phi_{i}&-\sin \phi_{i}\\ \cos\chi_{i}\sin\phi_{i}&-\sin\chi_{i}\sin\phi_{i}&\cos\phi_{i}\\ \sin\chi_{i}&\cos\chi_{i}&0\end{bmatrix} \),  
\[b= \begin{bmatrix}0&0&-g\end{bmatrix} ^{T}\text{and }u_{i}= \begin{bmatrix}u_{ xi}&u_{yi}&u_{zi}\end{bmatrix} ^{T}\text{ is the virtual control input described by}\]  
\[ \begin{cases}u_{xi}=a_{1i}\cos\chi_{i}\cos\phi_{i}-a_{2i}\sin\chi_{i}\cos\phi_{ i}-a_{3i}\sin\phi_{i}\\ u_{yi}=a_{1i}\cos\chi_{i}\sin\phi_{i}-a_{2i}\sin\chi_{i}\sin\phi_{i}+a_{3i}\cos \phi_{i}\\ u_{zi}=a_{1i}\sin\chi_{i}-a_{2i}\cos\chi_{i}-g\end{cases} , \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Aspect** & **Existing Works** & **The Article** \\ \hline Reconfiguration & Geometry [9; 16; 36] & Communication \\ \hline Swarm scale & Static [16; 34; 35] & Dynamic \\ \hline Leader election & Single-source information [16] & Multi-sources information \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The innovation of the article compared with existing works. where \(u_{xi}\), \(u_{yi}\), \(u_{zi}\) are the accelerations on the x-axis, the y-axis and the z-axis of the follower. Thus, the kinematics of the \(i\)th follower in the leader-follower multi-UAVs can be further represented as  
\[ \begin{cases}\dot{q}_{i}=p_{i}\\ \dot{p}_{i}=u_{i}\end{cases} ,i=1,\ldots,N.