# Distributed Attitude Synchronization for Spacecraft Formation Flying via Event-Triggered Control
## 3 Main Results
\begin{split}\frac{d\Big{\|}\tilde{\dot{\epsilon}}_{i}\Big{\|}}{ dt}&\leq\left\|\tilde{\dot{\dot{\epsilon}}}_{i}\right\|=\left\| \tilde{\dot{\epsilon}}_{i}-\tilde{\dot{\epsilon}}_{i}\right\|=\left\|\tilde{ \dot{\epsilon}}_{i}\right\|=\left\|\sum_{l=1}^{n}a_{ij}\big{(}\ddot{\sigma}_ {i}-\ddot{\sigma}_{j}\big{)}+\ddot{\sigma}_{ei}\right\|\\ &\leq n\big{(}\big{\|}\ddot{\sigma}_{i}\big{\|}+\big{\|}\ddot{ \sigma}_{j}\big{\|}\big{)}+\left\|\ddot{\sigma}_{i}\right\|+\left\|\ddot{ \sigma}_{d}\right\|\leq 2(n+1)\delta_{2}\end{split}  \tag{30}\]  
The derivative of \(\|\widetilde{\eta}_{i}\|\) is:  
\[ \begin{split}\frac{d\|\widetilde{\dot{\eta}}_{i}\|}{dt}& \leq\left\|\tilde{\dot{\dot{\eta}}}_{i}\right\|=\left\|\frac{d \big{(}\omega_{i}^{\times}\big{(}t_{k}^{i}\big{)}J_{i}\omega_{i}\big{(}t_{k}^ {i}\big{)}+J_{i}\dot{\omega}_{d}\big{(}t_{k}^{i}\big{)}\big{)}}{dt}-\frac{d \big{(}\omega_{i}^{\times}J_{i}\omega_{i}+J_{i}\dot{\omega}_{d}\big{)}}{dt} \right\|\\ &=\left\|\frac{d\big{(}\omega_{i}^{\times}I_{i}\omega_{i}+J_{i} \dot{\omega}_{d}\big{)}}{\big{\|}}\right\|\\ &=\left\|\dot{\omega}_{i}{}^{\times}I_{i}\omega_{i}+\omega_{i}{}^ {\times}J_{i}\dot{\omega}_{i}+J_{i}\ddot{\omega}_{d}\right\|\\ &\leq\left\|\dot{\omega}_{i}{}^{\times}I_{i}\omega_{i}\right\|+ \left\|\omega_{i}{}^{\times}I_{i}\dot{\omega}_{i}\right\|+\left\|J\ddot{\omega }_{d}\right\|\\ &\leq 2\|I_{i}\|\|\omega_{i}\|\|\dot{\omega}_{i}\|+\|I_{i}\|\| \ddot{\omega}_{d}\|\\ &\leq 2\|I_{i}\|\delta_{1}\delta_{2}+\|I_{i}\|\delta_{3}\end{split}  \tag{31}\]  
In accordance with (29)-(31), we obtain:  
\[ \begin{split}\frac{d\|\widetilde{\dot{\chi}}_{i}\|}{dt}+\frac{d \Big{\|}\tilde{\dot{\dot{\epsilon}}}_{i}\Big{\|}}{dt}+\frac{d\|\widetilde{ \dot{\eta}}_{i}\|}{dt}&\leq 2(n+1)r\delta_{1}+2\delta_{2}+2(n+1) \delta_{2}+2\|I_{i}\|\delta_{1}\delta_{2}+\|I_{i}\|\delta_{3}\\ &\leq 2(n+1)r\delta_{1}+2(n+2)\delta_{2}+(2\delta_{1}\delta_{2}+ \delta_{3})I_{\max}\end{split}  \tag{32}\]  
Define \(\psi=2(n+1)r\delta_{1}+2(n+2)\delta_{2}+(2\delta_{1}\delta_{2}+\delta_{3})J_{ \max}\). Integrating (32) with initial conditions \(\widetilde{\dot{\chi}}_{i}\big{(}t_{k}^{i}\big{)}=0\), \(\tilde{\dot{\epsilon}}_{i}\big{(}t_{k}^{i}\big{)}=0\), and \(\widetilde{\dot{\eta}}_{i}\big{(}t_{k}^{i}\big{)}=0\) leads to:  
\[\left\|\widetilde{\dot{\chi}}_{i}\right\|+\left\|\tilde{\dot{\dot{\epsilon}}}_{ i}\right\|+\left\|\widetilde{\dot{\eta}}_{i}\right\|\leq\int_{t_{k}^{i}}^{t}\psi=\psi \Big{(}t-t_{k}^{i}\Big{)} \tag{33}\]  
For all \(t\in\left[t_{k}^{i},t_{k+1}^{i}\right)\), \(\left\|\widetilde{\dot{\chi}}_{i}\right\|+\left\|\tilde{\dot{\dot{\epsilon}}}_{ i}\right\|+\left\|\tilde{\dot{\eta}}_{i}\right\|\) grows from zero to \(\frac{\sqrt{2}}{2}\|\dot{\chi}_{i}\|+c_{1}e^{-c_{2}\tanh(t)}\). At the next triggering time \(t=t_{k+1}^{i}\), (33) can be rearranged as:  
\[\left\|\widetilde{\dot{\chi}}_{i}\right\|+\left\|\tilde{\dot{\dot{\epsilon}}}_{ i}\right\|+\left\|\widetilde{\dot{\eta}}_{i}\right\|\leq\psi T_{i} \tag{34}\]  
Based on the above analysis, we can get:  
\[T_{i}\geq\frac{\sqrt{2}\|\dot{\chi}_{i}\|}{2\psi}+\frac{c_{1}e^{-c_{2}\tanh(t) }}{\psi} \tag{35}\]  
According to (35), we can set \(\tau_{i}^{\min}\) as \(\frac{c_{1}e^{-\frac{c_{2}\pi}{2}}}{\psi}\), which is strictly positive, indicating that the SFF system is Zeno-free. This completes the proof of Theorem 2. \(\square\)