## 4 Simulations
### Case without the Radial Control  
The deputy's initial state is \(\mathbf{X}_{I}(t_{0})= \begin{bmatrix}0&1000&0&0.5&0&1\end{bmatrix} ^{\mathsf{T}}\), the terminal state of the deputy is \(\mathbf{X}_{I}(t_{f})= \begin{bmatrix}0&500&0&0.3&0&0.5\end{bmatrix} ^{\mathsf{T}}\), given that \(\mathsf{J}2\) perturbation is the main disturbance in LEO. A nonlinear dynamical model of \(\mathsf{J}2\)-perturbed spacecraft relative motion is introduced to evaluate the robustness of the system [27]. Meanwhile, another periodic disturbance is also incorporated into the dynamical model expressed as:  
\[\mathbf{D}=D_{m} \begin{bmatrix}\sin(n_{\mathsf{C}}t)\\ \sin(n_{\mathsf{C}}t)\\ \cos(n_{\mathsf{C}}t)\end{bmatrix} , \tag{41}\]  
where \(D_{m}\) is a constant. Furthermore, the linear sliding mode controller (LSMC) was introduced for comparative analysis [10]. For the case without radial thrust, the LSMC is designed as:  
\[\mathbf{U}_{L1}=-m_{0}((A_{13}e_{1u}+A_{14}e_{1a})+\mathbf{P}_{12}(A_{11}e_{1u}+A_{12} e_{1a})+c_{1}(\mathbf{P}_{12}e_{1u}+e_{1a})+K_{11}\mathbf{s}_{L1}+K_{12}\mathrm{ sig}^{\sigma_{1}}(\mathbf{s}_{L1})) \tag{42}\]  
The sliding mode surface is designed as \(\mathbf{s}_{L1}=\mathbf{e}_{1a}+\mathbf{P}_{12}\mathbf{e}_{1u}+c_{1}\overline{\mathbf{e}}_{1u}\), where \(c_{1}>0\) is the controller parameter. Furthermore, \(k_{11}\), \(k_{12}\) and \(k_{13}\), are selected according to ideas given in [13]. The other control parameters for SNNC and LSMC are shown in Table 2. To facilitate comparison between the two types of controllers, the controller parameters are selected on the principle that similar control energy is consumed to complete the formation reconfiguration mission. A comparison of the relative position errors and velocity errors of the SNNC and LSMC are shown in Figures 2 and 3. Details of the trajectories of relative position errors and relative velocity errors since \(\mathsf{T}\) is also enlarged in the right side of Figures 2 and 3, respectively, from which it can be seen that the steady-state relative position error for both \begin{table}
\begin{tabular}{c c} \hline
**Controller** & **Parameter** \\ \hline \multirow{5}{*}{SNNC} & \(a_{1}\!\!=\!2\times 10^{-3},\gamma_{1}=1,k_{11}=0.99\sqrt{2}|k_{12}|\), \\  & \(k_{12}=-1.95\sqrt{\omega_{C}},k_{13}=1/\sqrt{\omega_{C}},\mathbf{K}_{1}=diag(2 \times 10^{-3},3\times 10^{-3})\), \\  & \(\mathbf{\eta}_{1}=diag(10^{-5},10^{-6}),\mathbf{\xi}_{1}=diag(8.4\times 10^{-5},8.4\times 10^{-5})\), \\  & \(\mathbf{\xi}_{2}=diag(4\times 10^{-4},8\times 10^{-6})\), \\  & \(\mathbf{\xi}_{3}=diag(1.5\times 10^{-6},1\times 10^{-7})\), \(a=6,b=50\), \(\sigma_{1}=0.5\), \\  & \(b_{j}=1.2\times 10^{3}\) \\ \multirow{2}{*}{LSMC} & \(a_{1}\!\!=\!2\times 10^{-3},k_{11}=0.99\sqrt{2}|k_{12}|,k_{12}=-1.95\sqrt{\omega_{C}},k_{13}= 1/\sqrt{\omega_{C}},\\  & \(K_{11}=10^{-3}\cdot diag(2,3),\mathbf{K}_{12}=10^{-6}\cdot diag(10,1),\sigma_{1}= 0.5\) \\ \hline \end{tabular}
\end{table} 
Table 2: Closed-loop controller parameters without radial control. types of controllers is in the order of \(10^{0}\) m. The steady-state error is smaller and the control accuracy is higher with the SNNC compared to the LSMC controller. Comparisons of the SNNC and LSMC control input are depicted in Figure 4, with a magnitude of \(10^{-3}\) m/s\({}^{2}\) and approximately 0.8 period to reach steady state. The reconfiguration trajectory of the spacecraft is shown in Figure 5, which shows that the SNNC is able to perform the formation reconfiguration mission and verifies the feasibility of formation reconfiguration in the absence of radial thrust. Figure 3: Time histories of relative velocity errors for the case without radial control. Figure 2: Time histories of relative position errors for the case without radial control. To further compare the performance of the two types of controllers, the following evaluation indicators are defined.