# Adaptive Fuzzy-Based Fault-Tolerant Control of a Continuum Robotic System for Maximly Sinus Surgery
## 2 Surgical Robot Modeling
### Fuzzy Auto Regressive with Exogenous Input (ARX) Laguerre System Modeling  
Based on Figure 2, the proposed system modeling has three main steps: (a) ARX technique, (b) improve the robustness of the ARX technique by the orthonormal method and design of the ARX-Laguerre method, and (c) improve the accuracy of system estimation based on the fuzzy ARX-Laguerre technique. An extensive variety of sinus surgery continuum robots based on the ARX method is represented by the following equation [37]:  
\[ \begin{array}{l}W(k)+\lambda_{W_{1}}W(k-1)+\lambda_{W_{2}}W(k-2)+\ldots+ \lambda_{W_{W}}W(k-n_{W})=\lambda_{H_{1}}H(k-1)+\lambda_{H_{2}}H(k-2)\\ +\ldots+\lambda_{H_{H}}H(k-n_{H})+e(k)\end{array} . \tag{6}\]  
Therefore, the future output can be predicted by observation parameters based on Equation (7). \[ \begin{array}{l}W(k)=-\lambda_{W_{1}}W(k-1)-\lambda_{W_{2}}W(k-2)-\ldots- \lambda_{W_{W}}W(k-n_{W})+\lambda_{H_{1}}H(k-1)+\lambda_{H_{2}}H(k-2)\\ +\ldots+\lambda_{H_{H}}H(k-n_{H})+e(k)\end{array} , \tag{7}\]  
where \((W(k-n_{W}),H(k-n_{H})),W(k),H(k),(\lambda_{W},\lambda_{H})\), and \(e(k)\) are system lag, output, input, coefficients, and zero-mean noise, respectively. Building on Equation 7, the state-space ARX modeling is represented by Equation (8). \[\left\{ \begin{array}{l}W(k)=-\varepsilon(k).\psi+e(k)\\ \psi=\left[\begin{array}{cccc}\lambda_{W_{1}}\ldots&\lambda_{W_{W}}&\lambda_ {H_{1}}\ldots&\lambda_{H_{H}}\end{array}\right]\\ \varepsilon=\left[\begin{array}{cccc}-W(k-1)\ldots&-W(k-n_{W})&H(k-1)\ldots &H(k-n_{H})\end{array}\right]\end{array} .\right. \tag{8}\]  
Here, \(\varepsilon\) and \(\psi\) are regressor variables of the ARX system modeling and the coefficient matrix, respectively. Thus, Equation (8) represents the ARX estimation technique. \[ \begin{array}{l}W(k)=\hat{\phi}[W(k-1),W(k-2),\ldots,W(k-n_{W}),H(k-1),H(k-2 ),\ldots,H(k-n_{H})],\end{array}  \tag{9}\]  
\[ \begin{array}{l}\hat{\phi}=\arg\min_{\hat{\phi}}[f(\hat{\phi},\psi)\\ f(\hat{\phi},\psi)=\frac{1}{k}\sum\limits_{0}^{k}\left(W(k)-\varepsilon(k) \psi\right)^{2}\end{array} , \tag{10}\]  
Therefore, the state-space ARX system modeling is represented by the following equation. \[\left\{ \begin{array}{l}X_{q,F_{\varepsilon}\tau}(k+1)=\alpha_{X}X_{q,F_{ \varepsilon}\tau}+\alpha_{U}U_{F_{\varepsilon}\tau}(k)+\delta(k)\\ Y_{q}(k+1)=(\kappa)^{T}X_{q,F_{\varepsilon}\tau}\end{array} \right.. \tag{11}\]  
Here, \((\alpha_{X},\alpha_{U})\) are coefficients. To improve the accuracy and reduce the system estimation order, the ARX-Laguerre technique for modeling the surgical robot for the sinus is represented by the following equation [18; 21]. \[ \begin{array}{l}Y_{q}(k)=\sum\limits_{0}^{i_{V}}\alpha_{n,Y}(\sum\limits_{j =1}^{\infty}\lambda_{Y}*Y_{q}(k)).\chi_{n,Y}(k)+\sum\limits_{0}^{i_{U}}\alpha_{ n,U}(\sum\limits_{j=1}^{\infty}\lambda_{U}*U(k)).\chi_{n,U}(k),\end{array}  \tag{12}\]where \((\alpha_{n,Y}\&\alpha_{n,U}),(i_{Y},i_{U}),(\lambda_{Y},\lambda_{U}),\star,\chi_{n,Y (k)}\) and \(\chi_{n,U}(k)\) are the Fourier coefficients, the robot manipulator order, the function of the Laguerre orthonormal, the product of the convolution, the filter system output, and the filter system input, respectively. Figure 7 illustrates the block diagram of the ARX-Laguerre algorithm. Here, \(\phi_{0_{1}}=\frac{\sqrt{1-\varepsilon_{d}}}{Z-\zeta_{d}}\), \(\phi_{0_{2}}=\phi_{0_{n}}=\frac{1-\zeta_{d}z}{Z-\zeta_{d}}\), \(\phi_{i_{1}}=\frac{\sqrt{1-\zeta_{d}}}{Z-\zeta_{d}}\), \(\phi_{i_{2}}=\phi_{i_{n}}=\frac{1-\zeta_{d}z}{Z-\zeta_{d}}\), \(\kappa=[G_{0,a},\ldots,G_{n,a},G_{0,b},\ldots,G_{n,b}]\), and \((\zeta_{d},\zeta_{b})\) is the orthonormal basis. Therefore, the state-space ARX-Laguerre surgical robot estimation is represented by the following equation. \[\left\{ \begin{array}{l}X_{q,F_{\tau}\tau}(k+1)=[\alpha_{X}X_{q,F_{\tau}\tau}+ \alpha_{U}U_{F_{\tau}\tau}(k)+\alpha_{Y}Y_{q}(k)]+\delta(k)\\ Y_{q}(k+1)=(\kappa)^{T}X_{q,F_{\tau}\tau}\end{array}