## 4 Singular Perturbation Decomposition of the System  
The flexible space robot after capturing the satellite is a complex nonlinear rigid-flexible coupling system. Using the singular perturbation method, the system is decomposed into a slow subsystem including a base, joint motion, and link vibration. The fast subsystem involves base vibration and joint vibration. The control torque of the joint motor is designed as follows:  
\[\mathbf{\tau}_{m}=(\mathbf{I}+\mathbf{K}_{c})\mathbf{\tau}_{n}-\mathbf{K}_{c}\mathbf{\tau} \tag{17}\]  
where \(\mathbf{K}_{c}\) is a joint flexibility compensator and \(\mathbf{\tau}_{n}\) is composed of a slow sub-controller \(\mathbf{\tau}_{ns}\) and a fast sub-controller \(\mathbf{\tau}_{nf}\). Let \(\mathbf{N}=\left[ \begin{array}{cc}\mathbf{N}_{11}&\mathbf{N}_{12}\\ \mathbf{N}_{21}&\mathbf{N}_{22}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{D}_{M11 }&\mathbf{D}_{M12}\\ \mathbf{D}_{M21}&\mathbf{D}_{M22}\end{array} \right]^{-1}\). We define \(\mathrm{T}_{f}=\mathrm{T}/\sqrt{\mu}\) as the fast-changing timescale, and \(\mu=1/\min(k_{\rm b},k_{\rm m1},k_{\rm m2})\) as the singular perturbation factor. The state variables of the system under a fast timescale are \(\mathbf{q}_{f}=\left[q_{f1},q_{f2},\mathbf{q}_{f3}^{\rm T},\mathbf{q}_{f4}^{\rm T}\right]^{ \rm T}\), where \(q_{f1}=x_{bf}-\overline{x}_{bf}\), \(q_{f2}=\sqrt{\mu x}_{bf}\), \(\mathbf{q}_{f3}=\sigma_{f}-\overline{\sigma}_{f}\), and \(\mathbf{q}_{f4}=\sqrt{\mu}\sigma_{f}\). \(\overline{\ast}\) is a new expression of \(\ast\) when \(\mu\) tends to zero, and \(\ast_{f}\) is a new variable of \(\ast\) under a fast timescale. From the above variables and Equations (14)-(17), the slow subsystem of the system is obtained as follows:  
\[\left[ \begin{array}{cc}\mathbf{Q}_{11}&\mathbf{Q}_{12}\\ \mathbf{Q}_{21}&\mathbf{Q}_{22}\end{array} \right]\left[ \begin{array}{c}\ddot{\mathbf{q}} \\ \ddot{\mathbf{\delta}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{R}_{11}&\mathbf{R}_{ 12}\\ \mathbf{R}_{21}&\mathbf{R}_{22}\end{array} \right]\left[ \begin{array}{c}\dot{\mathbf{q}}\\ \dot{\mathbf{\delta}}\end{array} \right]+\left[ \begin{array}{c}\mathbf{0}_{2\times 1 }\\ \mathbf{0}_{2\times 1}\\ \mathbf{K}_{\rm\delta}\end{array} \right]=\left[ \begin{array}{c}\tau_{0}\\ \mathbf{\tau}_{\rm ns}\\ \mathbf{0}_{4\times 1}\end{array} \right] \tag{18}\]where \(\mathbf{Q}_{11}\in\mathbf{R}^{3\times 3}\), \(\mathbf{Q}_{12}\in\mathbf{R}^{3\times 4}\), \(\mathbf{Q}_{21}\in\mathbf{R}^{4\times 3}\), \(\mathbf{Q}_{22}\in\mathbf{R}^{4\times 4}\), \(\mathbf{R}_{11}\in\mathbf{R}^{3\times 3}\), \(\mathbf{R}_{12}\in\mathbf{R}^{3\times 4}\), \(\mathbf{R}_{21}\in\mathbf{R}^{3\times 4}\), \(\mathbf{R}_{21}\in\mathbf{R}^{4\times 3}\), \(\mathbf{R}_{22}\in\mathbf{R}^{4\times 4}\), \(\left[ \begin{array}{cc}\mathbf{Q}_{11}&\mathbf{Q}_{12}\\ \mathbf{Q}_{21}&\mathbf{Q}_{22}\end{array} \right]=\overline{\mathbf{D}}_{M22}+ \left[ \begin{array}{cc}0&\mathbf{0}_{1\times 2}&\mathbf{0}_{1\times 4}\\ \mathbf{0}_{2\times 1}&(\mathbf{I}+\mathbf{K}_{\mathrm{c}})^{-1}\mathbf{J}_{m}& \mathbf{0}_{2\times 4}\\ \mathbf{0}_{4\times 1}&\mathbf{0}_{4\times 2}&\mathbf{0}_{4\times 4}\end{array}  \right]\), and  
\[\left[ \begin{array}{cc}\mathbf{R}_{11}&\mathbf{R}_{12}\\ \mathbf{R}_{21}&\mathbf{R}_{22}\end{array} \right]=\overline{\mathbf{H}}_{M22}.\]  
The fast subsystem of the system is as follows:  
\[\mathrm{d}q_{\mathrm{f}}/\mathrm{d}t=A_{\mathrm{f}}q_{\mathrm{f}}+B_{\mathrm{f} }\tau_{\mathrm{nf}} \tag{19}\]  
where \(B_{\mathrm{f}}=\left[\mathbf{0}_{4\times 2}^{\mathrm{T}},\left[J_{m}^{-1}( \mathbf{I}+\mathbf{K}_{\mathrm{c}})\right]^{\mathrm{T}}\right]^{\mathrm{T}}\) and \(A_{\mathrm{f}}=\left[