# Distributed Attitude Synchronization for Spacecraft Formation Flying via Event-Triggered Control
## 3 Main Results  
In this section, an attitude cooperative control algorithm with event-triggered strategy for SFF is proposed such that all spacecraft in formation achieve attitude synchronization. Before proceeding, the following two auxiliary variables are introduced, given by:  
\[e_{i}=\sum_{j=1}^{n}a_{ij}\big{(}\sigma_{i}-\sigma_{j}\big{)}+\sigma_{ei} \tag{6}\]  
\[\chi_{i}=\omega_{ei}+re_{i} \tag{7}\]where \(r>0\) is constant and \(a_{ij}\) denotes the element of the weighted adjacency matrix \(A\). \(r>0\) is to make \(\sigma_{ei}\) converge to a small bound of the origin after \(\chi_{i}\) approaches zero. Based on (3) and (5)-(7), we obtain:  
\[ \begin{split} J_{i}\dot{\chi}_{i}&=J_{i}(\dot{ \omega}_{i}-\dot{\omega}_{d})+rJ_{i}\dot{e}_{i}\\ &=u_{i}-\omega_{i}{}^{\times}J_{i}\omega_{i}-J_{i}\dot{\omega}_{d }+rJ_{i}\dot{e}_{i}\end{split}  \tag{8}\]  
For \(i\)th spacecraft, \(t_{k}^{i}\) represents the \(k\)th (\(k=1,2,3,\cdots\)) triggering time instant. The sampling values of \(e_{i}\) and \(\chi_{i}\) are denoted by \(\tilde{e}_{i}\) and \(\hat{\chi}_{i}\), which are defined, respectively, as:  
\[\dot{e}_{i}=e_{i}\left(t_{k}^{i}\right) \tag{9}\]  
\[\hat{\chi}_{i}=\omega_{ei}\left(t_{k}^{i}\right)+r\tilde{e}_{i} \tag{10}\]  
for all \(t\in\left[t_{k}^{i},t_{k+1}^{i}\right)\), where \(t_{k+1}^{i}\) denotes the next trigger moment. A distributed event-based attitude coordinated controller is established for SFF with limited energy and computation resources, which is given by:  
\[u_{i}(t)=-k_{i}\hat{\chi}_{i}-rJ_{i}\dot{e}_{i}+\omega_{i}^{\times}\left(t_{k }^{i}\right)J_{i}\omega_{i}\left(t_{k}^{i}\right)+J_{i}\dot{\omega}_{d}\left( t_{k}^{i}\right) \tag{11}\]  
where \(k_{i}=k_{0}+r\|J_{i}\|+1\) and \(k_{0}\) is a positive constant. Then, define the estimation errors of \(\dot{e}_{i}\), \(\chi_{i}\), and \(\omega_{i}{}^{\times}J_{i}\omega_{i}+J_{i}\dot{\omega}_{d}\) as \(\hat{\widetilde{e}}_{i}\), \(\widetilde{\chi}_{i}\), and \(\widetilde{\eta}_{i}\), respectively. The estimation errors are given by:  
\[\dot{\widetilde{e}}_{i}=\dot{\hat{e}}_{i}-\dot{e}_{i} \tag{12}\]  
\[\widetilde{\chi}_{i}=\hat{\chi}_{i}-\chi_{i} \tag{13}\]  
\[\widetilde{\eta}_{i}=\omega_{i}^{\times}\left(t_{k}^{i}\right)J_{i}\omega_{i} \left(t_{k}^{i}\right)+J_{i}\dot{\omega}_{d}\left(t_{k}^{i}\right)-\omega_{i} {}^{\times}J_{i}\omega_{i}-J_{i}\dot{\omega}_{d} \tag{14}\]  
Subsequently, the event-triggered strategy of \(i\)th spacecraft is designed as:  
\[t_{k+1}^{i}=\text{inf}\Bigg{\{}t>t_{k}^{i}\Bigg{|}\|\widetilde{\chi}_{i}\|+ \left\|\ddot{\widetilde{e}}_{i}\right\|+\|\widetilde{\eta}_{i}\|\geq\frac{ \sqrt{2}}{2}\|\hat{\chi}_{i}\|+c_{1}e^{-c_{2}\text{tanh}(i)}\Bigg{\}} \tag{15}\]  
where \(c_{1}>0\) and \(c_{2}>0\) are constants. \(c_{1}\) and \(c_{2}\) are designed as positive constants such that \(c_{1}e^{-c_{2}\text{tanh}(i)}\) keeps monotone positive decreasing, which will make the upper bounds of the system states errors smaller and smaller over time. **Remark 3**. : _Under the developed event-triggered strategy (15), the latest state information about its neighbors is only required in the calculation of the event-triggered function, which means that the calculation of the event-triggered strategy requires less energy and computing resources. In addition, the controller is only updated at its own triggering instants, which greatly reduces energy consumption and computational burden. When the event-triggered strategy (15) is satisfied, each spacecraft broadcasts the state information to its adjacent spacecraft and obtains state information about its adjacent spacecraft to make \(\|\widetilde{\chi}_{i}\|+\left\|\ddot{\widetilde{e}}_{i}\right\|+\|\widetilde{ \eta}_{i}\|\) reset to zero._  
**Theorem 1**. : _Consider the SFF system described by (1)-(3), the event-driven cooperative control scheme is constructed according to (11) and the event-triggered function is designed as (15). Then, the system trajectories are able to reach the neighborhood near the origin._  
Proof.