## 4 Numerical Simulation  
### Simulation of Controller Performance  
In this section, we introduce a numerical simulation for reconfiguring a two-satellite electromagnetic formation. This simulation assesses the performance of our nonlinear relative motion dynamics model and the integrated control strategy. This example is motivated by the use of Satellite Formation Flying (SFF) in Synthetic Aperture Radar(SAR) missions, which require non-Keplerian orbits and continuous control to maintain the desired formation configuration [29]. We assume that the center of mass of the formation follows a circular orbit at an altitude of 500 km. The two satellites, which are identically configured, are designed to orbit the formation's center of mass with a rotation period of 2 h and a radius of 15 m. The mass of the satellite is 50 kg, and the magnetic dipole on SatB is selected with a constant value of \(3\times 10^{6}\) A \(\cdot\) m\({}^{2}\) point to SatA. The super-boundary magnitude of the magnetic dipole is \(5\times 10^{5}\) A \(\cdot\) m\({}^{2}\). The total external disturbance acceleration acting on the formation system is taken as \([6,-5-4]^{\text{T}}\times 10^{-6}\)m/s\({}^{2}\)[30]. The initial relative state error is chosen as  
\[\mathbf{e}_{r}=[-0.5,0.2,0.6]^{\text{T}}\text{m},\dot{\mathbf{e}}_{r}=[0.06,0.01,-0.04]^ {\text{T}}\text{m}/\text{s}\]  
The mass center orbit parameters and initial conditions are shown in Tables 1 and 2. The subscripts a and b represent SatA and B.  
The controller parameters are set as  
\[\tilde{\mathbf{R}}=\begin{bmatrix}6.42&0&0\\ 0&6.25&0\\ 0&0.0&6\end{bmatrix},\mathbf{K}_{j}=\begin{bmatrix}0.128&0&0\\ 0&0.12&0\\ 0&0&0.12\end{bmatrix},\mathbf{K}_{w}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\]  
For comparison, some simulation results from [30] are cited in the result figures with their control strategy \(\mathbf{u}_{ADRC}\). Figure 2 illustrates the ideal relative trajectory and the trajectories of SatA/B under the control of \(\mathbf{u}_{DRPBC}\), respectively. Subsequently, Figure 3 presents the variation in relative error between the actual trajectory and the relative orbit, while Figure 4 depicts the variation of the magnetic dipole on SatA. As observed in the figures, employing the DRPBC method results in the trajectory error reducing to below 1 mm within 150 s, with the eventual error consistently maintained below \(10^{-4}\) m. The strength of the magnetic dipoles remains within the super-boundary limits. It is evident that the DRPBC method exhibits significant advantages in terms of error convergence speed and control accuracy.  
\begin{table}
\begin{tabular}{c c} \hline
**Item** & **Values** \\ \hline Radius \(q_{0}\) [km] & 6878.137 \\ Eccentricity \(e\) & 0 \\ Inclination \(i\) [deg] & 60 \\ Argument of perigee \(\omega\) [deg] & 90 \\ Right ascension of ascending node \(\Omega\) [deg] & 0 \\ True anomaly \(v\) [deg] & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of mass center orbits.  
\begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Values** \\ \hline \(q_{a}(0)\) [m] & \(\begin{bmatrix}12.7950&1.4490&7.6934\end{bmatrix}^{\text{T}}\) \\ \(q_{a}(0)\) [m/s] & \(\begin{bmatrix}0.002&0.0117&-0.0055\end{bmatrix}^{\text{T}}\) \\ \(q_{b}(0)\) [m] & \(\begin{bmatrix}-12.7950&-1.4490&-7.6934\end{bmatrix}^{\text{T}}\) \\ \(q_{b}(0)\) [m/s] & \(\begin{bmatrix}0.002&0.0117&-0.0055\end{bmatrix}^{\text{T}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Initial condition of simulations.  
Figure 4: Magnetic dipole on SatA.  
Figure 3: Relative position error.  
Figure 2: Satellite trajectory.