## 3 Model Decomposition of Fully Flexible Base-Link-Joint Space Robot
\begin{array}{cc}0&\mathbf{0}_{1\times 2}&\mathbf{0}_{1\times 4}\\ \mathbf{0}_{2\times 1}&(I+\mathbf{K}_{\rm c})^{-1}\mathbf{I}_{\rm m}&\mathbf{0}_{2\times 4}\\ \mathbf{0}_{4\times 1}&\mathbf{0}_{4\times 2}&\mathbf{0}_{4\times 4}\end{array} \right],\mathbf{S} \Big{(}\mathbf{q},\delta,\dot{\mathbf{q}},\dot{\mathbf{\delta}}\Big{)}=\overline{\mathbf{H}}_{ \rm aa}\mathbf{\tau}_{\rm qj}=\left[ \begin{array}{c}\mathbf{\tau}_{0},\mathbf{\tau}_{ \rm ns}^{T}\end{array} \right]^{T},\]  
they are all submatrices corresponding to \(\mathbf{N}(q_{\rm b},\mathbf{q},\delta)\). Define \(\wp\) to be an arbitrary matrix, then \(\overline{\wp}\) to be a new expression for \(\wp\) when \(\mu\to 0\). Due to the coupling effect between the rigid motion of the slow-variable subsystem and the flexible vibration of the rod, which affects the operating accuracy. For this reason, before designing the trajectory tracking control algorithm, the slow subsystem needs to be decoupled, the \(\mathbf{R}\) and \(\mathbf{S}\) in Equation (17) are written in separate blocks as \(\left[ \begin{array}{cc}R_{11}&\mathbf{R}_{12}\\ \mathbf{R}_{21}&\mathbf{R}_{22}\end{array} \right]\) and \(\left[ \begin{array}{cc}S_{11}&S_{12}\\ S_{21}&S_{22}\end{array} \right]\), where \(\mathbf{R}_{11},\mathbf{S}_{11}\in\Re^{3\times 3}\), \(\mathbf{R}_{12},\mathbf{S}_{12}\in\Re^{3\times 4}\), \(\mathbf{R}_{21},\mathbf{S}_{21}\in\Re^{4\times 3}\), \(\mathbf{R}_{22},\mathbf{S}_{22}\in\Re^{4\times 4}\). Considering the external disturbance, the dynamic model of the fully driven rigid subsystem is obtained by decoupling Equation (17) as follows:  
\[\mathbf{M}(\delta,\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}\Big{(}\delta,\dot{\delta},\mathbf{q}, \dot{\mathbf{q}}\Big{)}\dot{\mathbf{q}}+\mathbf{c}_{1}\Big{(}\delta,\dot{\delta},\mathbf{q}, \dot{\mathbf{q}}\Big{)}=\mathbf{\tau}_{\rm oj}+\mathbf{d}, \tag{18}\]  
where \(\mathbf{M}(\delta,\mathbf{q})=\mathbf{R}_{11}-\mathbf{R}_{12}\mathbf{R}_{22}^{-1}\mathbf{R}_{21}\), \(\mathbf{C}\Big{(}\delta,\dot{\delta},\dot{\mathbf{q}},\dot{\mathbf{q}}\Big{)}=\mathbf{S}_{11}-\bm {R}_{12}\mathbf{R}_{22}^{-1}\mathbf{S}_{21}\) is the column vector of centrifugal force and Coriolis force equivalent to the rigid subsystem, \(\mathbf{c}_{1}\Big{(}\delta,\dot{\delta},\mathbf{q},\dot{\mathbf{q}}\Big{)}=\Big{(}\mathbf{S}_{12} -\mathbf{R}_{12}\mathbf{R}_{22}^{-1}\mathbf{S}_{22}\Big{)}\dot{\delta}-\mathbf{R}_{12}\mathbf{R}_{ 22}^{-1}\mathbf{K}_{\rm\delta}\dot{\delta}\) is the nonlinear term of system dynamics, \(\mathbf{d}\) is the external interference applied to the driving motor of the fully flexible space robot. According to the principle of mutual independence of fast and slow time scales, the corresponding subsystem is found based on the fast time scale. Take fast time scale as \(t_{\rm f}=t/\sqrt{\mu}\), fast subsystem state variable as \(\mathbf{q}_{\rm f}=\left[q_{\rm f1},q_{\rm f2},\mathbf{q}_{\rm f3}^{\sf T},\mathbf{q}_{\rm f4 }^{\sf T}\right]^{\sf T}\). \(q_{\rm f1}=q_{\rm bf}-\overline{q}_{\rm bf}\), \(q_{\rm f2}=\sqrt{\mu}q_{\rm bf}\), \(q_{\rm f3}=\sigma_{\rm f}-\overline{\sigma}_{\rm f}\), \(q_{\rm f4}=\sqrt{\mu}\tilde{\sigma}_{\rm f}\). The flexible deformation of the base and joint under the fast time scale are \(q_{\rm bf}\) and \(\sigma_{\rm f}\) respectively. Take the derivative of \(\mathbf{q}_{\rm f}\), the dynamics equation of the fast subsystem is as follows:  
\[d\mathbf{q}_{\rm f}/dt_{\rm f}=A_{\rm f}\mathbf{q}_{\rm f}+B_{\rm f}\mathbf{\tau}_{\rm nf}, \tag{19}\]  
\[\text{where}\,\mathbf{A}_{\rm f}=\left[ \begin{array}{ccc}0&1&\mathbf{0}_{1\times 2}& \mathbf{0}_{1\times 2}\\ -\overline{N}_{11}k_{\rm bf}&0&\mathbf{N}_{12}^{*}\mathbf{K}_{\rm mf}&\mathbf{0}_{1\times 2 }\\ \mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 2}&\mathbf{I}_{2\times 2}\\ \mathbf{0}_{2\times 1}&\mathbf{0}_{2\times 1}&-\mathbf{J}_{\rm m}^{-1}(\mathbf{I}+\mathbf{K}_{\rm c}) \mathbf{K}_{\rm mf}&\mathbf{0}_{2\times 2}\end{array}