## 3 Model Decomposition of Fully Flexible Base-Link-Joint Space Robot  
The outstanding difficulty of active control of motion vibration of fully flexible space robots lies in the existence of triple flexibility, and active suppression of the flexibility of one class of components will cause excitation of another class, which is difficult to control. To solve the above problem, the system flexibility is analyzed at different time scales using the singular perturbation method. The total controller of motor with joint flexibility compensation is designed as:  
\[\mathbf{\tau}_{\rm m}=(\mathbf{I}+\mathbf{K}_{\rm c})\mathbf{\tau}_{\rm n}-\mathbf{K}_{\rm c}\mathbf{ \tau}, \tag{12}\]where \(\mathbf{I}\in\Re^{2\times 2}\) is the unit matrix, \(\mathbf{K}_{\rm c}\in\Re^{2\times 2}\) is the symmetric positive definite flexible compensation matrix, \(\mathbf{\tau}_{\rm n}\in\Re^{2}\) is the controller to be designed, and the expression is:  
\[\mathbf{\tau}_{\rm n}=\mathbf{\tau}_{\rm ns}+\mathbf{\tau}_{\rm nf}, \tag{13}\]  
where \(\mathbf{\tau}_{\rm ns}\in\Re^{2}\) is the slow subsystem controller, \(\mathbf{\tau}_{\rm nf}\in\Re^{2}\) is the fast-variable subsystem controller. Substituting Equation (12) into Equation (10), we have:  
\[\ddot{\mathbf{\sigma}}=\mathbf{I}_{\rm m}^{-1}(\mathbf{I}+\mathbf{K}_{\rm c})(\mathbf{\tau}_{\rm n }-\mathbf{\tau})-\ddot{\mathbf{q}}_{\rm j}. \tag{14}\]  
Equation (9) is written in blocks as:  
\[\left[ \begin{array}{cc}\mathbf{D}_{\rm bb}&\mathbf{D}_{\rm ba}\\ \mathbf{D}_{\rm ab}&\mathbf{D}_{\rm aa}\end{array} \right]\left[ \begin{array}{c}\ddot {q}_{\rm b}\\ \dot{\mathbf{q}}\\ \dot{\mathbf{\delta}}\end{array} \right]+\left[ \begin{array}{cc}H_{\rm bb}&\mathbf{ H}_{\rm ba}\\ H_{\rm ab}&\mathbf{H}_{\rm aa}\end{array} \right]\left[ \begin{array}{c}\dot{q}_{ \rm b}\\ \dot{\mathbf{q}}\\ \dot{\mathbf{\delta}}\end{array} \right]+\left[ \begin{array}{c}k_{\rm b}q_{\rm b }\\ 0\\ -\mathbf{\tau}\\ \mathbf{K}_{\rm\delta}\end{array} \right]=\left[ \begin{array}{c}0\\ \mathbf{\tau}_{0}\\ \mathbf{0}_{2\times 1}\\ \mathbf{0}_{4\times 1}\end{array} \right], \tag{15}\]  
where \(\mathbf{D}_{\rm bb}\),\(\mathbf{H}_{\rm bb}\in\Re^{1\times 1}\), \(\mathbf{D}_{\rm aa}\), \(\mathbf{H}_{\rm ba}\in\Re^{1\times 7}\mathbf{D}_{\rm ab}\),\(\mathbf{H}_{\rm ab}\in\Re^{7\times 1}\), \(\mathbf{D}_{\rm aa}\),\(\mathbf{H}_{\rm aa}\in\Re^{7\times 7}\), they are all submatrices corresponding to \(\mathbf{D}(q_{\rm b},\mathbf{q},\delta)\) and \(\mathbf{H}(q_{\rm b},\mathbf{q},\delta,\dot{q}_{\rm b},\dot{q},\dot{\delta})\). Define the relation equation as follows:  
\[\mathbf{N}=\left[ \begin{array}{cc}N_{11}&N_{12}\\ N_{21}&N_{22}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{D}_{\rm bb}&\bm {D}_{\rm ba}\\ \mathbf{D}_{\rm ab}&\mathbf{D}_{\rm aa}\end{array} \right]^{-1}, \tag{16}\]  
where \(\mathbf{N}_{11}\in\Re^{1\times 1}\), \(N_{12}\in\Re^{1\times 7}\), \(N_{21}\in\Re^{7\times 1}\), \(\mathbf{N}_{22}\in\Re^{7\times 7}\), they are all submatrices corresponding to \(\mathbf{N}(q_{\rm b},\mathbf{q},\delta)\). Let \(\mathbf{q}\), \(\delta\) be the slow sub-variant, \(q_{\rm b}\), \(\mathbf{\sigma}\) be the fast sub-variant, and \(\mu=1/\min(k_{\rm b},k_{\rm m1},k_{\rm m2})\) be the singular perturbation factor. Substituting Equation (16), \(\mathbf{\tau}=\mathbf{K}_{\rm m}\mathbf{\sigma}\) and \(\mu=0\) into Equation (15), the kinetic equation of the slow subsystem is obtained as follows:  
\[\mathbf{R}(\mathbf{q},\mathbf{\xi})\left[ \begin{array}{c}\ddot{\mathbf{q}}\\ \dot{\mathbf{\delta}}\end{array} \right]+\mathbf{S}\Big{(}\mathbf{q},\delta,\dot{\mathbf{q}}, \dot{\mathbf{\delta}}\Big{)}\left[ \begin{array}{c}\dot{\mathbf{q}}\\ \dot{\mathbf{\delta}}\end{array} \right]+\left[ \begin{array}{c}\mathbf{0}_{3\times 1} \\ \dot{\mathbf{K}}_{\rm\delta}\dot{\mathbf{\delta}}\end{array} \right]=\left[ \begin{array} []{c}\mathbf{\tau}_{\rm qj}\\ \mathbf{0}_{4\times 1}\end{array} \right], \tag{17}\]  
\[\text{where}\,\mathbf{R}(\mathbf{q},\mathbf{\xi})=\overline{\mathbf{D}}_{\rm aa}+\left[