# Adaptive Fuzzy-Based Fault-Tolerant Control of a Continuum Robotic System for Maximly Sinus Surgery
## 3 Proposed Method for Fault Estimation, Detection, Identification and Tolerant Control
### Fault-Tolerant Control
\begin{array}{l}\tau_{p}(k)=\tau_{onlineFLC}(k)+\hat{Y}_{q(p-online)}(k )\\ \tau_{onlineFLC}(k)=-I(q)\times(\hat{X}_{d}+\alpha_{P-new}e(k)+\alpha_{d-new} \dot{e}(k)+\alpha_{i-new}\dot{\sum e}(k))+f(\hat{X}_{q,F_{c}\tau(p)},U)\\ \hat{Y}_{q(p)}(k+1)=(\kappa)^{T}\hat{X}_{q,F_{c}\tau(p)}(k)\\ \hat{X}_{q,F_{c}\tau(p)}(k+1)=[\alpha_{X}\hat{X}_{q,F_{c}\tau(p)}+\alpha_{U}U_{ F_{c}\tau}(k)+\alpha_{Y}\hat{Y}_{q(p)}(k)+\alpha_{F}\hat{f}_{fuzzy}(p)]\\ +\delta_{(p)}(k)+\alpha_{P-new}(Y_{q}(k)-\hat{Y}_{q(p)}(k))+f(\hat{X}_{q,F_{c} \tau(p)},U)+I^{-1}(\alpha_{P-new}(Y_{q}(k)-\hat{Y}_{q(p)}(k)))\end{array} \right. \tag{35}\]  
Here, \(\tau_{p}(k),\tau_{onlineFLC}(k),\hat{Y}_{q(p-online)}(k)\), and \((\alpha_{P-new},\alpha_{d-new},\alpha_{i-new})\) are the proposed method for fault-tolerant control in the surgical robot for the sinus, the online tuning part of the feedback linearization controller, the online tuning part of the proposed fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization observer, and the online tuning coefficients, respectively. The online tuning proportional, derivative, and integral coefficients are represented by Equation (36). \[\left\{ \begin{array}{l}\alpha_{P-new}=\alpha_{P}\times\alpha_{online}\\ \alpha_{d-new}=\alpha_{d}\times\alpha_{online}\\ \alpha_{i-new}=\alpha_{i}\times\alpha_{online}\end{array} \right. \tag{36}\]  
where \(\alpha_{online}\) is fuzzy output for tuning the proportional, derivative, and integral coefficients. The fuzzy algorithm has two inputs (error and change of error) and one output. The fuzzy membership functions in the interval of \([-0.1,0.1]\) are Gaussian, and the fuzzy sets are defined as NB, NM, NS, Z, PS, PM, and PB. The fuzzy membership functions for change of error in the interval of \([-2,2]\) are Gaussian, and the fuzzy sets are defined as NB, NM, NS, Z, PS, PM, and PB. The fuzzy membership functions for fuzzy output in the interval of \([0.1,2]\) are Gaussian, and the fuzzy sets are defined as PS, PM, and PB. The rule table for online tuning the proposed fuzzy fault-tolerant controller is given in Table 4. Figure 13 illustrates the power of fault-tolerant control in the manipulator of the surgical robot for the sinus for online tuning of the fuzzy observer feedback linearization controller (proposed method), observer-based feedback linearization controller, and classical feedback linearization controller. Regarding the figure, the power of fault reduction in the proposed method is significantly better than with the other two methods. The proposed algorithm for fault diagnosis and the fault-tolerant controller of the surgical robot for the sinus are summarized in Algorithm 1. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & \multicolumn{6}{c}{**Change of Error (de)**} \\ \cline{3-10}  & & **NB** & **NM** & **NS** & **Z** & **PS** & **PM** & **PB** \\  & **NB** & PB & PB & PB & PB & PM & PS & PS \\  & **NM** & PB & PB & PB & PM & PS & PS & PS \\
**Eror (e)** & **NS** & PB & PB & PM & PS & PS & PS & PS \\  & **Z** & PB & PM & PS & PS & PS & PS & PS \\  & **PS** & PB & PB & PB & PB & PM & PS & PS \\  & **PM** & PB & PB & PM & PS & PS & PS \\  & **PB** & PB & PB & PM & PS & PS & PS \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fuzzy rule table for online tuning of the proposed fault-tolerant control algorithm.