## 3 Preliminaries
### Kinematic Analysis of the Continuum Robot  
The continuum manipulator studied in this paper consists of three segments. To maintain generality, assume a total of \(n\) segments, and here \(n=3\). The arc length parameter of the \(i\)-th \((i=1,2,\ldots,n)\) segment is and the absolute position at the axis center is  
\[\mathbf{r}_{i}(\cdot,s)=\mathbf{r}_{i-1}(\cdot,1)+\mathbf{R}_{i-1}\mathbf{R}_{z}(\varphi_{i}) \mathbf{p}(\theta_{i},s) \tag{6}\]  
where \(\cdot\) in \(\mathbf{r}_{i}(\cdot,s)\) represents other independent variable parameters. Subsequently, for the sake of symbol simplicity, only arc length parameters will be provided for independent variables, disregarding other variables that include the arc length parameters. For example, \(\mathbf{r}_{i}(\cdot,s)\) will be further abbreviated as \(\mathbf{r}_{i}(s)\). Also, \(\mathbf{r}_{0}(1)\) is equivalent to \(\mathbf{r}_{0}\), and similar expressions will not be elaborated in the following text. Considering the offset between the installation position of the manipulator and the base, the absolute position at the axis center for the first segment with an arc length parameter of \(s\) is  
\[\mathbf{r}_{1}(s)=\mathbf{r}_{0}+\mathbf{R}_{0}(\mathbf{b}+\mathbf{R}_{z}(\varphi_{1})\mathbf{p}( \theta_{1},s)) \tag{7}\]  
where \(\mathbf{b}\) represents the installation position on the base for the first segment of the continuum manipulator. This position corresponds to the coordinates of the intersection point between the base and the central axis of the first segment in the fixed coordinate system of the base. The rotation matrix representing the transformation from the local coordinate system at the section with an arc length parameter of \(s\) in the \(i\)-th segment to the inertial coordinate system is given by  
\[\mathbf{R}_{i}(s)=\mathbf{R}_{i-1}(1)\mathbf{R}_{z}(\varphi_{i})\mathbf{R}_{y}(s\theta_{i})\bm {R}_{z}(-\varphi_{i}) \tag{8}\]  
This matrix represents the orientation of the section relative to the inertial system. The absolute linear velocity at the axis center of the \(i\)-th segment with an arc length parameter of \(s\) is given by  
\[ \begin{split}\mathbf{v}_{i}(s)&=\mathbf{v}_{i-1}(1)+\mathbf{ \omega}_{i-1}(1)\times r_{i-1,i}(s)\\ &\quad+\mathbf{R}_{i-1}(1)\mathbf{R}_{z}(\varphi_{i})\mathbf{p}(s,\theta_{i}) \phi_{i}\\ &\quad+\mathbf{R}_{i-1}(1)\mathbf{R}_{z}(\varphi_{i})\mathbf{p}_{\theta_{i}}( \theta_{i},s)\theta_{i}\\ &=\mathbf{I}_{i,p}(s)\hat{\mathbf{q}}\end{split}  \tag{9}\]  
where \(\mathbf{r}_{i-1,i}(s)=\mathbf{r}_{i}(s)-\mathbf{r}_{i-1}(1)\) is the position vector from the end of the \((i-1)\)-th segment to the position in the \(i\)-th segment with an arc length parameter of \(s\), \(\times\) denotes the cross product, \(\mathbf{R}_{z_{i}\varphi_{i}}\) is \(\partial\mathbf{R}_{z}(\varphi_{i})/\partial\varphi_{i}\), \(\mathbf{p}_{\theta_{i}}(\theta_{i},s)\) is \(\partial\mathbf{p}(\theta_{i},s)/\partial\theta_{i}\), and \(\mathbf{J}_{i,p}(s)\) is the corresponding linear velocity Jacobian matrix. The absolute angular velocity at the axis center of the \(i\)-th segment with an arc length parameter of \(s\) is given by  
\[ \begin{split}\mathbf{\omega}_{i}(s)&=\mathbf{\omega}_{i-1} (1)\\ &\quad+\mathbf{R}_{i-1}\big{[}\mathbf{e}_{3}\phi_{i}+\mathbf{R}_{z}(\varphi_{i })\mathbf{e}_{2}s\theta_{i}-\mathbf{R}_{z}(\varphi_{i})\mathbf{R}_{y}(s\theta_{i})\mathbf{e}_ {3}\phi_{i}\big{]}\\ &=\mathbf{J}_{i,\omega}(s)\hat{\mathbf{q}}\end{split}  \tag{10}\]  
where \(\mathbf{J}_{i,\omega}(s)\) is the corresponding angular velocity Jacobian matrix. The linear momentum of the support disk for the \(i\)-th segment of the continuum manipulator is given by  
\[\mathbf{P}_{i,d}=\sum_{j=1}^{n_{1}}m_{d}\mathbf{v}_{i}\left(\,j/n_{1}\right) \tag{11}\]  
where \(m_{d}\) is the mass of the disk and \(j\) is the \(j\)-th support disk on the continuum manipulator, where \(j\) ranges from \(1\) to the total number \(n_{1}\) of support disks on the \(i\)-th segment of the continuum manipulator.