## 3 Preliminaries
### Dynamic Model for Tracking Control  
The previous section has already investigated the kinetic and potential energies of the base and the continuum manipulator. Lagrangian formalism can be employed to perform dynamic modeling of the continuum space robot system. The total kinetic energy of the system is  
\[K=K_{0,t}+K_{0,r}+\sum_{i=1}^{3}\left(K_{i,dt}+K_{i,dt}+K_{i,rt}+K_{i,rr}\right) \tag{42}\]  
The total potential energy of the system is  
\[U=\frac{1}{2l}\sum_{i=1}^{3}EI_{xx}\theta_{i}^{2} \tag{43}\]  
In the space environment, where the gravitational potential energy is effectively zero, the total potential energy here only considers the elastic potential energy of the continuum manipulator. The Lagrangian function for the system is  
\[L=K-U \tag{44}\]  
The [PERSON]'s equations for the system are  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\mathbf{\dot{q}}}\right)- \frac{\partial L}{\partial\mathbf{\dot{q}}} =Q \tag{45}\] \[=\frac{d}{dt}\left(\frac{\partial K}{\partial\mathbf{\dot{q}}}\right) -\frac{\partial K}{\partial\mathbf{\dot{q}}}+\frac{\partial U}{\partial\mathbf{\dot{q}}}\]  
where \(\mathbf{Q}\) is the generalized forces  
Substituting (42) and (43) into (45), the dynamic model of the system is  
\[M(\mathbf{\dot{q}})\mathbf{\dot{q}}+\mathbb{C}(\mathbf{\dot{q}},\mathbf{\dot{q}})\mathbf{\dot{q}}+ N\mathbf{\dot{q}}=\mathbf{\tau} \tag{46}\]  
where \(M(\mathbf{\dot{q}})\in\mathbf{R}^{12\times 12}\) is the inertia matrix of the continuum space robot system, \(\mathbb{C}(\mathbf{\dot{q}},\mathbf{\dot{q}})\in\mathbf{R}^{12\times 12}\) is the matrix representing centrifugal and Coriolis forces, \(\mathbf{N}\in\mathbf{R}^{12\times 12}\) is the generalized stiffness matrix, and \(\bar{\tau}\in R^{12\times 1}\) is the generalized force. These symbols can be expressed as  
\[M=M_{0t}+M_{0r}+\sum_{i=1}^{3}\left(M_{i,dt}+M_{i,dt}+M_{i,rt}+M_{i,rr}\right) \tag{47}\]  
\[C\hat{\mathbf{q}} =\hat{M}\hat{\mathbf{q}}-\frac{\partial\mathbf{K}}{\partial\bar{\mathbf{q}}}\] \[=\hat{M}\hat{\mathbf{q}}-\frac{1}{2} \begin{bmatrix}\bar{\mathbf{q}}^{ \mathsf{T}}\frac{\partial\mathbf{M}}{\partial\bar{\mathbf{q}}_{1}}\hat{\mathbf{q}}\\ \vdots\\ \hat{\mathbf{q}}^{\mathsf{T}}\frac{\partial\mathbf{M}}{\partial\bar{\mathbf{q}}_{1}}\hat{ \mathbf{q}}\end{bmatrix}  \tag{48}\] \[\mathbf{N}\bar{\mathbf{q}} =\frac{\partial\mathbf{U}}{\partial\bar{\mathbf{q}}}=\frac{EI_{xx}}{l^{2 }}[0_{7\times 1}\;\theta_{1}\;\theta_{2}\;0\;\theta_{3}]^{\mathsf{T}} \tag{49}\]  
Representing the equations of motion shown in (46) in a block-wise form, it follows that  
\[ \begin{bmatrix}M_{00}&M_{0q}\\ M_{q0}&M_{qq}\end{bmatrix}\begin{bmatrix}\bar{q}_{0}\\ \bar{q}\end{bmatrix} + \begin{bmatrix}C_{0}\\ C_{q}\end{bmatrix} + \begin{bmatrix}N_{0}\\ N_{q}\end{bmatrix} = \begin{bmatrix}\tau_{0}\\ \tau\end{bmatrix}  \tag{50}\]  
where  
\[ \begin{bmatrix}C_{0}\\ C_{q}\end{bmatrix} =C\hat{\mathbf{q}}, \begin{bmatrix}N_{0}\\ N_{q}\end{bmatrix} =\mathbf{N}\bar{\mathbf{q}}, \begin{bmatrix}\tau_{0}\\ \tau\end{bmatrix} =\bar{\mathbf{\tau}} \tag{51}\]  
where \(\bar{\mathbf{\tau}}\) represents the generalized force vector, including input torques and their equivalent generalized torques due to forces or torques at each joint, \(\tau_{0}\) is the \(6\times 1\) generalized force of the base, and \(\tau\) is the \(6\times 1\) generalized force of each joint in the continuum manipulator [6]. Due to the free-floating nature of the base, the corresponding control quantity is zero. The above equation is equivalent to  
\[ \begin{split}&(-M_{q0}M_{00}^{-1}M_{0q}+M_{qq})\bar{\mathbf{q}}\\ &\quad+M_{q0}M_{00}^{-1}(-N_{0}-C_{0})+N_{q}+C_{q}=\tau\end{split}  \tag{52}\]  
Simplifying, it follows that  
\[A\bar{q}+B=\mathbf{u} \tag{53}\]  
where \(\mathbf{A}=-M_{q0}M_{00}^{-1}M_{0q}+M_{qq}\), \(\mathbf{B}=M_{q0}M_{00}^{-1}(-N_{0}-C_{0})+N_{q}+C_{q}\), and \(\mathbf{u}=\tau\) is the control input.