# Distributed Attitude Synchronization for Spacecraft Formation Flying via Event-Triggered Control
## 3 Main Results
\begin{split}\dot{V}&\leq-\sum_{i=1}^{n}k_{0}\|\dot{x} _{i}\|^{2}-\sum_{i=1}^{n}k_{i}\|\dot{\chi}_{i}\|^{2}+\sum_{i=1}^{n}k_{i}\bigg{(} \frac{\sqrt{2}}{2}\|\dot{\chi}_{i}\|+c_{1}e^{-c_{2}\tanh(t)}\bigg{)}^{2}\\ &\leq-\sum_{i=1}^{n}k_{0}\|\dot{x}_{i}\|^{2}-\sum_{i=1}^{n}k_{i}\| \dot{\chi}_{i}\|^{2}+2\sum_{i=1}^{n}k_{i}\left[\left(\frac{\sqrt{2}}{2}\|\dot{ \chi}_{i}\|\right)^{2}+\left(c_{1}e^{-c_{2}\tanh(t)}\right)^{2}\right]\\ &=-\sum_{i=1}^{n}k_{0}\|\dot{x}_{i}\|^{2}-\sum_{i=1}^{n}k_{i}\| \dot{\chi}_{i}\|^{2}+\sum_{i=1}^{n}k_{i}\bigg{[}\|\dot{\chi}_{i}\|^{2}+2 \Big{(}c_{1}e^{-c_{2}\tanh(t)}\Big{)}^{2}\bigg{]}\\ &=-\sum_{i=1}^{n}k_{0}\|\dot{\chi}_{i}\|^{2}+2\Big{(}c_{1}e^{-c_ {2}\tanh(t)}\Big{)}^{2}\sum_{i=1}^{n}k_{i}\end{split}  \tag{25}\]  
Define \(k_{\max}=\max(k_{1},k_{2},\cdots,k_{n})\) and \(J_{\max}=\max(\|J_{1}\|,\|J_{2}\|,\cdots,\|J_{n}\|)\). The above equation can be rewritten as:  
\[ \begin{split}\dot{V}&\leq-\frac{k_{0}}{j_{\max}} \sum_{i=1}^{n}J_{\max}\|\dot{\chi}_{i}\|^{2}+2n\Big{(}c_{1}e^{-c_{2}\tanh(t)} \Big{)}^{2}k_{\max}\\ &\leq-\mu V+\Delta\end{split}  \tag{26}\]  
where \(\mu=k_{0}/J_{\max}\) and \(\Delta=2n\Big{(}c_{1}e^{-c_{2}\tanh(t)}\Big{)}^{2}k_{\max}\). Integrating (26) gives:  
\[V(t)\leq\bigg{(}V(0)-\frac{\Delta}{\mu}\bigg{)}e^{-\mu t}+\frac{\Delta}{\mu} \tag{27}\]  
From (27), we can conclude that the state errors of the SFF system are able to converge to the desired neighborhood under the proposed controller (11) with the developed event-triggered mechanism (15). The proof of Theorem 1 is completed.\(\square\)  
**Remark 4**. : _It is important to realize that the design parameters \(c_{1}\) and \(c_{2}\) influence the inter-event time and the control performance simultaneously because the trigger threshold \(c_{1}e^{-c_{2}\tanh(t)}\) depends on \(c_{1}\) and \(c_{2}\)[29]. For one thing, the increase of \(c_{1}\) and the decrease of \(c_{2}\) are beneficial to the increase of the inter-event time, which means that the amount of computation will be reduced. For another, small \(c_{1}\) and large \(c_{2}\) will lead to good control performance, shrinking the convergence errors._  
For ith spacecraft, set the inter-event time among two consecutive events as \(T_{i}=t_{k+1}^{i}-t_{k}^{i}\) and the minimum inter-event time (MIET) as \(\tau_{i}^{\min}\), in which \(\tau_{i}^{\min}\leq T_{i}\). In the following, the possibility of Zeno behavior can be excluded rigorously by proving that the MIET is positive. **Theorem 2**. : _For the SFF system (1)-(3) with the event-based attitude coordination control strategy (11) and the event-triggered mechanism (15), if Assumptions 1-4 hold true, the MIET \(\tau_{i}^{\min}\) will be strictly positive, showing that the system has no Zeno phenomenon._  
**Proof. ** Taking the derivative of \(\|\widetilde{\dot{x}}_{i}\|\) gives:  
\[\frac{d\|\widetilde{\dot{x}}_{i}\|}{dt}\leq\left\|\widetilde{\dot{x}}_{i}\right\| =\left\|\dot{\dot{x}}_{i}-\dot{x}_{i}\right\|=\left\|\dot{\dot{x}}_{i}\right\| \tag{28}\]According to (6) and (7), (28) can be rewritten as:  
\[ \begin{split}\frac{d\|\tilde{\chi}_{i}\|}{dt}&\leq\left\| \dot{\omega}_{ei}+r\left[\sum_{l=1}^{n}a_{ij}\big{(}\dot{\sigma}_{i}-\dot{ \sigma}_{j}\big{)}+\dot{\sigma}_{ei}\right]\right\|\\ &\leq\left\|\dot{\omega}_{i}\right\|+\left\|\dot{\omega}_{a} \right\|+nr\big{\|}\dot{\sigma}_{i}\big{\|}+nr\big{\|}\dot{\sigma}_{j}\big{\|}+ r\big{\|}\dot{\sigma}_{i}\big{\|}+r\big{\|}\dot{\sigma}_{d}\big{\|}\\ &\leq 2(n+1)r\delta_{1}+2\delta_{2}\end{split}  \tag{29}\]  
Similarly, the derivative of \(\left\|\tilde{\dot{\epsilon}}_{i}\right\|\) is:  
\[