## 3 Preliminaries  
### Deformation Description of the Continuum Robot  
Common methods for describing the deformation of the continuum manipulator include the constant curvature method, finite element method, and Cosserat rod theory. To simplify the modeling process and focus on the control of the continuum manipulator, the constant curvature method is chosen as the foundation for kinematic modeling. According to the PCC assumption, a continuous robot can be approximated as a series of subsegments with uniform curvature.  
For a segment of a continuum manipulator with a length of \(l\), as illustrated in Figure 4, its motion deformation can be described as a section of rod with constant curvature [18].  
Define the rotational transformation as:  
\[R_{z}(\varphi_{i})=\begin{bmatrix}c_{\varphi_{i}}&-s_{\varphi_{i}}&0\\ s_{\varphi_{i}}&c_{\varphi_{i}}&0\\ 0&0&1\end{bmatrix},R_{y}(\theta_{i})=\begin{bmatrix}c_{\theta_{i}}&0&s_{ \theta_{i}}\\ 0&1&0\\ -s_{\theta_{i}}&0&c_{\theta_{i}}\end{bmatrix} \tag{1}\]  
where \(c\) represents \(\cos\) and \(s\) represents \(\sin\).  
Define the angle between the plane containing a single segment of the continuum manipulator's arc and the plane \(xOz\) as the curvature plane angle \(\varphi\), as illustrated in Figure 5. The \(i\)-th segment of the continuum manipulator, when \(\varphi_{i}=0\), is depicted in Figure 6.  
Figure 4: Continuum manipulator bending diagram.  
In Figure 6, \(\theta_{i}\) is the angle formed by \(x_{i}\) and \(x_{i-1}\), \(\theta_{i}\) is defined as the bending angle for the \(i\)-th segment, and \(r_{i}\) is the bending radius for the \(i\)-th segment. From Figure 6, the local position of the end of this segment after bending can be obtained.  
\[\begin{split}\mathbf{r}_{i,lcl}&=[(1-c_{\theta_{i}}) \ 0\ s_{\theta_{i}}]^{\mathsf{T}}r_{i}\\ &=[(1-c_{\theta_{i}})\ 0\ s_{\theta_{i}}]^{\mathsf{T}}\frac{l}{ \theta_{i}}\end{split} \tag{2}\]  
where the subscript \({}_{lcl}\) represents in the local coordinate system. When \(\mathbf{r}_{i,lcl}=\begin{bmatrix}0&0&l\end{bmatrix}^{\mathsf{T}}\).  
To obtain the pose of every point on the continuum manipulator, it is proposed to use the arc length parameter for modeling. Consider a segment of the continuum manipulator as an arc, define the arc length parameter \(s\in[0,\ 1]\), where \(s=0\) corresponds to the beginning of the arc and \(s=1\) corresponds to the end of the arc. When \(\theta_{i}=0\),  
Figure 5: Bending plane angle \(\varphi\).  
Figure 6: Coordinate transformation when \(\varphi_{i}=0\).  
\(\mathbf{r}_{i,lcl}=\begin{bmatrix}0&0&s\end{bmatrix}^{T}\). Under the PCC assumption, the position coordinates of any point in the arm segment can be expressed as  
\[\begin{split}\mathbf{r}_{i,lcl}(\theta_{i},s)&=\begin{bmatrix}(1-c_{s \theta_{i}})&0&s_{s\theta_{i}}\end{bmatrix}^{T}\frac{l}{\theta_{i}}\\ &\triangleq\mathbf{p}(\theta_{i},s)\end{split} \tag{3}\]  
When the curvature plane angle \(\varphi_{i}\
eq 0\), the position of any point in the \(i\)-th segment of the robotic arm in the local coordinates of the \(i\)-th segment can be represented as  
\[\mathbf{r}_{i,lcl}(\varphi_{i},\theta_{i},s)=\mathbf{R}_{z}(\varphi_{i})\mathbf{p}(\theta _{i},s) \tag{4}\]  
Attitude can be represented by a rotation transformation matrix [19] as  
\[\mathbf{R}_{i,lcl}(\varphi_{i},\theta_{i},s)=\mathbf{R}_{z}(\varphi_{i})\mathbf{R}_{y}(s \theta_{i})\mathbf{R}_{z}(-\varphi_{i}) \tag{5}\]  
In practical applications, adjusting the flexibility of a continuum manipulator and adapting to various complex scenarios can be achieved by concatenating several subsegments, enhancing the flexibility at the robot system.