# Adaptive Fuzzy-Based Fault-Tolerant Control of a Continuum Robotic System for Maximly Sinus Surgery
## 2 Surgical Robot Modeling
### Surgical Robot Kinematics and Dynamics  
As shown in Figure 3, a surgical robot for the sinus has two segments and two deflection stages (DS). Figure 4 shows that the proximal segment (PS) consists of areas II and III, which include seven joints, and the distal segment comprises 10 joints for areas I and IV. To analyze the continuum surgical robot for sinus deflection shown in Figure 4, the D-H conventions are presented in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Part** & **Quantity** & **Description** \\ \hline Manipulator & Degrees of freedom & 4 \\ Continuum module & Diameter & 4 [mm] \\  & Number of joints & 17 \\  & Maximum bending angle & 270\({}^{\circ}\) \\  & Length of continuum module & 30 [mm] \\  & Diameter of holes & 0.6 [mm] \\  & Minimum radius of curvature & 5.8 [mm] \\  & Radius of curvature at 180\({}^{\circ}\) & 10 [mm] \\ Grippers & Length & 10 [mm] \\ Cables (gripper) & Diameter & 0.2 [mm] \\  & Material & Stainless Steel \\ Cables (deflection) & Diameter & 0.36 [mm] \\  & Material & Stainless Steel \\ NiTiNo1 tube & Outer diameter & 0.508 [mm] \\  & Inner diameter & 0.305 [mm] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of a continuum sinus robot [3]. Figure 3: Continuum module of a surgical robot for the sinus: (**a**) continuum joint, (**b**) motion schematic, and (**c**) structure. The translation matrix is defined by the following equation:  
\[\begin{split} \begin{split}\begin{subarray}{c}i-1\\ i\end{subarray}T=\left[\begin{array}{cccc}\cos\theta_{i}&-\sin\theta_{i}&0&a_{i-1} \\ \sin\theta_{i}\cos\alpha_{i-1}&\cos\theta_{i}\cos\alpha_{i-1}&-\sin\alpha_{i-1} &-\sin\alpha_{i-1}d_{i}\\ \sin\theta_{i}\sin\alpha_{i-1}&\cos\theta_{i}\sin\alpha_{i-1}&\cos\alpha_{i-1 }&\cos\alpha_{i-1}d_{i}\\ 0&0&0&1\end{array}\right]\end{split}  \tag{1}\]  
Here, \( \begin{subarray}{c}i-1\\ i\end{subarray} T,a_{i},\alpha_{i},d_{i}\) and \(\theta_{i}\) are the transformation matrix for each link, the distance from \(\hat{Z}_{i}\) to \(\hat{Z}_{i+1}\) measured along \(\hat{X}_{i}\), the angle from \(\hat{Z}_{i}\) to \(\hat{Z}_{i+1}\) measured about \(\hat{X}_{i}\), the distance from \(\hat{X}_{i-1}\) to \(\hat{X}_{i}\) measured along \(\hat{Z}_{i}\), and the angle from \(\hat{X}_{i-1}\) to \(\hat{X}_{i}\) measured about \(\hat{Z}_{i}\), respectively. Based on Equation (1), the forward kinematics is represented by Equation (2). \[ \begin{split} FK=\begin{subarray}{c}0\\ 1\end{subarray}T=\begin{subarray}{c}0\\ 1\end{subarray}T=\begin{subarray}{c}0\\ 1\end{subarray}T=\begin{subarray}{c}T2\\ 1\end{subarray}T2\begin{subarray}{c}3\\ 3\end{subarray}T4\begin{subarray}{c}4\\ 5\end{subarray}T5\begin{subarray}{c}5\\ 6\end{subarray}T7\end{subarray}T8\begin{subarray}{c}7\\ 9\end{subarray}T9\begin{subarray}{c}10\\ 10\end{subarray}T1\begin{subarray}{c}110\\ 11\end{subarray}T12\begin{subarray}{c}13\end{subarray}T14\begin{subarray}{c} 147\\ 15\end{subarray}T16\end{subarray}T17\end{split}  \tag{2}\]  
A slave robot manipulator to drive a surgical continuum robot has 4-DOF, is serial linked, and is a coupling effects system. The configurations for continuum sinus surgery with the master robot are  
Figure 4: Four areas of deflection for sinus surgical continuum robot. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Link** & \(\mathbf{\alpha}\) & **a** & **d** & \(\mathbf{\theta}\) & **State** \\ \hline
**1** & 0 & \(a_{1}\) & \(d_{1}\) & 0 & \(\theta\leq 90^{\circ}\) \\
**2** & \(-90^{\circ}\) & 0 & \(d_{2}  
illustrated in Figure 5. As shown in Figure 5, the master manipulator is used for input control, and four controllers are designed to control the four motors located in the slave manipulator. As shown in Figure 6, these motors are used to actuate the translation, rotation, deflection, and gripper of the surgical robot for the sinus. The dynamic formulation of the surgical robot for the sinus is represented by Equation (3) [34; 35; 36].