## 4 Numerical Examples  
In this section, we present the practical implementation of the algorithm based on the aforementioned design, demonstrated through a specific numerical simulation example. A inner-formation background with specific parameters is given, and its dynamics is with the form of (1). The simulation parameters are selected as follows: the angular velocity vector of the out-spacecraft, denoted as \(n^{T}=\left[\begin{array}{cc}2&5&3\\ \end{array}\right]\times 10^{-3}\) rad/s, the mass of the out-spacecraft \(m_{out}=250\) kg, and the mass of the in-spacecraft \(m_{in}=1.5\) kg. Here, \(m_{out}\) and \(m_{in}\) correspond to \(m_{1}\) and \(m_{2}\) in (1), respectively. The external disturbance is considered \(\Delta a=\left[\begin{array}{cc}0.001\sin(2t+\frac{\pi}{4})&-0.003\sin(t+ \frac{\pi}{8})&0.002\sin(4t+\frac{\pi}{3})\end{array}\right]^{T}\!\!N\). Considering the distinct operational scenarios encountered by each axis in practical spacecraft systems, diverse axis parameters were selected herein, and the disturbances experienced during simulation of each axis also vary. Consequently, these disparities may give rise to variations in the simulation specifics of the three-axis system due to the influence of nonlinear dynamics.  
Based on these conditions, we propose a parameter turning in detail here. The parameter of the event-triggered observer is considered to be \(\ell_{1}=\ell_{2}=\ell_{3}=6\), \(K_{i}(\beta)=\left[\begin{array}{cc}1.2252&2.2267&0.7633\\ \end{array}\right]^{T}\!\!N\).  
**Remark 8**.: _Here, \(K_{i}(\beta)\) is obtained by condition (4). The calculation method can be obtained through feature root configuration methods or appropriate LMI (Linear Matrix Inequality) calculation methods. In this paper, we present an LMI-based approach for calculating the method. Specifically, we consider the following LMI conditions: \(Q_{1}^{T}P+PQ_{1}-Q_{2}^{T}R^{T}-RQ_{2}<0\), where_  
\[Q_{1}=\left[\begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\\ \end{array}\right]\!\!,\,Q_{2}=\left[\begin{array}{cc}1&0&0\\ \end{array}\right]\!\!,\,P\text{ is a positive definite matrix, and }R\text{ is a matrix of}\]  
_appropriate order. Calculating \(P\) and \(R\) through the LMI condition, we obtain \(K_{i}(\beta)=P^{-1}R\)._  
And a normal output feedback control is considered here. We use  
\[u_{1}(t)=-\frac{1}{50}\left[\begin{array}{ccc}5.2&5&-2\\ \end{array}\right]\!\!\left[\begin{array}{cc}\xi_{1}&\psi_{1}&\phi_{1}\\ \end{array}\right]\]  
\[u_{2}(t)=-\frac{1}{50}\left[\begin{array}{ccc}2.5&3.5&0\\ \end{array}\right]\!\!\left[\begin{array}{ccc}\xi_{1}&\psi_{1}&\phi_{1}\\ \end{array}\right]\]  
\[u_{3}(t)=-\frac{1}{50}\left[\begin{array}{ccc}6&5&-2\\ \end{array}\right]\!\!\left[\begin{array}{ccc}\xi_{1}&\psi_{1}&\phi_{1}\\ \end{array}\right]\]  
It is obvious to find that the controllers meet the conditions (20)-(22).  
The control results are depicted in Figure 1. As evident from the figure, employing an event-driven sampling design, the system converges to a specific range under the output feedback control rate.  
The transmission of sampling information is illustrated in Figure 2. It can be observed that during the initial stages of the system, when additional data are required to support control instructions, sampling occurs at a high frequency. However, once the system achieves stability, there is a significant decrease in the sampling rate, thereby indicating the rationality of this resource allocation design.  
The estimation results of the three axes in Figures 3-5 demonstrate convergence patterns consistent with the theoretical findings. By ensuring accurate estimations of displacement and velocity, the system disturbances and uncertainties are also estimated within a certain range.  
Figure 1: Control results for relative motion.  
Figure 3: Estimation results for X\(-\)axis, including the estimated results of displacement, velocity and acceleration.  
Figure 2: Transmission of sampling information. including the event-driven condition triggering in the X, Y, and Z axes.