## 3 Cross-Scale Decoupling of Manipulator System
### Second Singular Perturbation Decomposition  
The FSS (19) will be decoupled again. We can have the SSS characterizing the large overall motion of the manipulator without the whole flexibilities, and the SFS is only about the flexible vibration of the third link. We define the inverse matrix \(D\) of matrix \(M\) as  
\[D\ =\ M^{-1}\ =\ \begin{bmatrix}M_{1}M_{2}\\ M_{3}M_{4}\end{bmatrix} = \begin{bmatrix}D_{1}D_{2}\\ D_{3}D_{4}\end{bmatrix}  \tag{22}\]  
where \(M_{1}\in R^{3\times 3}\), \(D_{1}\in R^{3\times 3}\). Additionally, we make  
\[G(\theta,\dot{\theta},q,\dot{q})\ =\ \begin{bmatrix}G_{1}\\ G_{2}\end{bmatrix}  \tag{23}\]  
Substituting (22) and (23) into the first stage of the slow subsystem (19), we can obtain  
\[ \begin{array}{l}\ddot{\theta}=&-D_{2}(\theta,q)K_{x}q-D_{1}(\theta,q)G_{1} (\theta,\dot{\theta},q,\dot{q})-D_{2}(\theta,q)G_{2}(\theta,\dot{\theta},q, \dot{q})\\ &+D_{1}(\theta,q)\Big{(}-\widetilde{A}^{-1}\widetilde{B}\dot{\theta}+ \widetilde{A}^{-1}\widetilde{C}I_{s1}\Big{)}\end{array}  \tag{24}\]  
\[ \begin{array}{l}\ddot{q}=&-D_{4}(\theta,q)K_{x}q-D_{3}(\theta,q)G_{1}( \theta,\dot{\theta},q,\dot{q})-D_{4}(\theta,q)G_{2}(\theta,\dot{\theta},q, \dot{q})\\ &+D_{3}(\theta,q)\Big{(}-\widetilde{A}^{-1}\widetilde{B}\dot{\theta}+ \widetilde{A}^{-1}\widetilde{C}I_{s1}\Big{)}\end{array}  \tag{25}\]  
where \(K_{x}= \begin{bmatrix}k_{1}\\ k_{2}\end{bmatrix} \). We define the second singular perturbation parameter \(\varepsilon_{2}\ =\ \frac{1}{k}\) with \(k\ =\ \min(k_{1},k_{2})\). Comparing with the first singular perturbation parameter, they satisfy the following inequality: \(0<\varepsilon_{1}<<\varepsilon_{2}<<1\). We denote  
\[ \begin{array}{l}\widetilde{K}\ =\ \varepsilon_{2}K_{x}\\ q\ =\ \varepsilon_{2}y\end{array}  \tag{26}\]Setting \(\epsilon_{2}\ =\ 0\), substituting (26) into the (24) and (25), we can derive  
\[ \begin{split}\tilde{\theta}=&-D_{2,2 }(\theta,0)\tilde{k}_{y2}-D_{1,\Delta}(\theta,0)G_{1,\Delta 2}(\theta,\dot{\theta},0,0)-D_{2,2}(\theta,0)G_{2,\Delta 2 }(\theta,\dot{\theta},0,0)\\ +& D_{1,2}(\theta,0)\left(-\widetilde{A}^{-1} \tilde{B}\tilde{\theta}+\widetilde{A}^{-1}\tilde{C}I_{\Delta 2}\right)\\ 0=&-D_{4,2}(\theta,0)\tilde{k}_{y2}-D_{3,2}( \theta,0)G_{1,\Delta 2}(\theta,\dot{\theta},0,0)-D_{4,2}(\theta,0)G_{2,\Delta 2 }(\theta,\dot{\theta},0,0)\\ +& D_{3,2}(\theta,0)\left(-\widetilde{A}^{-1} \tilde{B}\tilde{\theta}+\widetilde{A}^{-1}\tilde{C}I_{\Delta 2}\right)\end{split}  \tag{27}\]  
By solving Equation (28), we obtain  
\[ \begin{split} y_{\Delta 2}=&-\widetilde{K}^{-1}D_{4, \Delta 2}^{-1}(\theta,0)D_{3,2}(\theta,0)G_{1,\Delta 2}(\theta,\dot{\theta},0,0)-\widetilde{K}^{-1}D_{4, \Delta 2}^{-1}(\theta,0)D_{4,\Delta 2}(\theta,0)G_{2,\Delta 2}(\theta,\dot{\theta},0,0)\\ +&\widetilde{K}^{-1}D_{4,2}^{-1}(\theta,0)D_{3,2}( \theta,0)\left(-\widetilde{A}^{-1}\tilde{B}\tilde{\theta}+\widetilde{A}^{-1} \tilde{C}I_{\Delta 2}\right)\end{split}  \tag{29}\]  
where subscript \(s2\) represents the second stage slow timescale, and \(I_{s2}\) is the control current under the second stage slow timescale. Substituting (29) into Equation (27), the SSS can be derived as  
\[M_{1,s2}(\theta,0)\tilde{\theta}+G_{1,s2}(\theta,\dot{\theta},0,0)\ =\ - \widetilde{A}^{-1}\tilde{B}\tilde{\theta}+\widetilde{A}^{-1}\widetilde{C}I_{s2} \tag{30}\]  
Similarly, ignoring \(\varrho(\epsilon_{2})\), the variables \(y\) and \(y\ =\ y_{f2}\ +\ y_{s2}\) can be represented as  
\[ \begin{split} y\ =\ y_{f2}\ +\ y_{s2}\\ I_{s1}\ =\ I_{f2}+I_{s2}\end{split}  \tag{31}\]  
where the subscript \(f2\) represents the second stage fast timescale. We introduce the second level of fast timescale \(\sigma_{2}\) in the boundary layer, and we have  
\[\sigma_{2}\ =\ \frac{\mathrm{t}}{\sqrt{\epsilon_{2}}} \tag{32}\]  
The second level slow variable \(y_{s2}\) is considered constant near the boundary layer \(\epsilon_{2}\to 0\).