## 2 Dynamic Modeling of Fully Flexible Base-Link-Joint Space Robot
### Fully Flexible Base-Link-Joint Space Robot Modeling  
According to the geometric position of the system in the inertial coordinate system, we have:  
\[\left\{\begin{array}{l}\mathbf{r}_{0}=\left(x_{0},y_{0}\right)^{T}\\ \mathbf{r}_{1}=\mathbf{r}_{0}+(l_{0}+q_{b})\mathbf{e}_{0}+x_{1}\mathbf{e}_{1}+  
Let \(\dot{\mathbf{r}}_{C}=0\), Equation (5) take the derivative, then calculate the kinetic energy of each split of the system as:  
\[\left\{\begin{array}{l}T_{\rm r0}=\frac{1}{2}m_{0}\dot{\tau}_{0}^{2}+\frac{1}{ 2}J_{0}\dot{\vartheta}_{0}^{2}\\ T_{\rm ri}=\frac{1}{2}\int_{0}^{l_{\rm t}}\rho_{\rm k}\dot{r}_{\rm d}\dot{\mathbf{x }}_{\rm k},(k=1,2)\\ T_{\rm mk}=\frac{1}{2}J_{\rm mk}\dot{\vartheta}_{mk}^{2},\ (k=1,2)\end{array}\right., \tag{6}\]  
where \(T_{\rm r0}\) indicates the kinetic energy of the base, \(T_{\rm rk}\) indicates the kinetic energy of link \(B_{\rm k}\), and \(T_{\rm mk}\) indicates the kinetic energy of the motor rotor at joint \(O_{\rm k}\).  
Neglecting the effect of weak gravity, the potential energy of each part of the system is as follows:  
\[\left\{\begin{array}{l}V_{\rm b}=\frac{1}{2}k_{\rm b}\vartheta_{\rm b}^{2} \\ V_{\rm ak}=\frac{1}{2}\sum\limits_{j=1}^{2}k_{\rm k}\delta\dot{\beta}_{kj^{ \prime}}^{2},(k=1,2)\\ V_{mk}=\frac{1}{2}k_{\rm mk}(q_{\rm mk}-q_{k})^{2},(k=1,2)\end{array}\right., \tag{7}\]  
where \(V_{\rm b}\) represents the elastic potential energy of the base, \(V_{\rm ak}\) represents the potential energy of link \(B_{k}\), \(k_{\rm k}\dot{\beta}_{kj}=EI_{k}\int_{0}^{l_{\rm k}}\bigg{(}\dot{\vartheta}_{ kj}^{2}(x_{k})\bigg{)}\mathrm{d}x_{k}\), \(V_{\rm mk}\) represent the potential energy of flexible joints \(O_{k}\).  
The Lagrange function is defined as:  
\[L=T_{\rm r0}+\sum\limits_{k=1}^{2}(T_{\rm rk}+T_{\rm mk})-V_{\rm b}-\sum \limits_{k=1}^{2}(V_{\rm ak}+V_{\rm mk}). \tag{8}\]  
Substituting Equation (8) into the Lagrange equation, the carrier position uncontrolled, attitude controlled fully flexible space robot dynamics model as follows:  
\[\mathbf{D}(q_{\rm b},\mathbf{q},\mathbf{\dot{\xi}})\left[\begin{array}{c}\ddot{q}_{\rm b }\\ \ddot{\mathbf{q}}\\ \ddot{\mathbf{\xi}}\end{array}\right]+\mathbf{h}\Big{(}q_{\rm b},\mathbf{q},\mathbf{\dot{\xi} },\dot{q}_{\rm b},\dot{\mathbf{q}},\dot{\mathbf{\xi}}\Big{)}+\left[\begin{array}{c}k _{\rm b}q_{\rm b}\\ 0\\ -\mathbf{\tau}\\ \mathbf{K}_{\rm\delta}\end{array}\right]=\left[\begin{array}{c}0\\ \tau_{0}\\ \mathbf{0}_{2\times 1}\\ \mathbf{0}_{4\times 1}\end{array}\right], \tag{9}\]  
\[\mathbf{I}_{\rm m}\ddot{q}_{\rm m}+\mathbf{\tau}=\mathbf{\tau}_{\rm m}, \tag{10}\]  
\[\mathbf{\tau}=\mathbf{K}_{\rm m}\mathbf{\sigma}, \tag{11}\]  
where \(\mathbf{D}(q_{\rm b},\mathbf{q},\mathbf{\dot{\xi}})\in\mathbf{R}^{8\times 8}\) is the symmetric positive definite mass matrix, \(\mathbf{h}\Big{(}q_{\rm b},\mathbf{q},\mathbf{\dot{\xi}},\dot{q}_{\rm b},\dot{\mathbf{q}}, \dot{\mathbf{\xi}}\Big{)}\in\mathbf{R}^{8\times 1}\) is the column vector of centrifugal and Koch forces, which can be written as the product of \(\mathbf{H}\Big{(}q_{\rm b},\mathbf{q},\mathbf{\dot{\xi}},\dot{q}_{\rm b},\dot{\mathbf{q}}, \dot{\mathbf{\xi}}\Big{)}\in\mathbf{R}^{8\times 8}\) and \(\left[\dot{q}_{\rm b}^{\rm T},\dot{\mathbf{q}}^{\rm T},\dot{\mathbf{\xi}}^{\rm T}\right] ^{\rm T}\cdot\mathbf{q}=\left[q_{0},\mathbf{q}_{\rm j}^{\rm T}\right]^{\rm T}\) is the column vector of carrier attitude and joint angle, \(\mathbf{q}_{\rm j}=[q_{1},q_{2}]^{\rm T}\). \(\mathbf{q}_{\rm m}=[q_{\rm m1},q_{\rm m2}]^{\rm T}\), \(\mathbf{\delta}=[\delta_{11},\delta_{12},\delta_{21},\delta_{22}]^{\rm T}\), \(\mathbf{K}_{\rm\delta}=\mathrm{diag}(k_{\rm 811},k_{\rm 812},k_{\rm 821},k_{\rm 822})\) is the diagonal matrix of flexible rod stiffness coefficients, \(\mathbf{I}_{\rm m}=\mathrm{diag}(J_{\rm m1},J_{\rm m2})\), \(\mathbf{K}_{\rm m}=\mathrm{diag}(k_{\rm m1},k_{\rm m2})\), \(\mathbf{\sigma}=\mathbf{q}_{\rm m}-\mathbf{q}_{\rm j}\), \(\mathbf{\tau}_{\rm m}=[\tau_{\rm m1},\tau_{\rm m2}]^{\rm T}\) is the motor rotor control torque.