# 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  
A fault-tolerant controller is used to reduce or eliminate the effect of faults in the surgical robot for the sinus. As shown in Figure 2, the fault-tolerant control section has two main subparts: (i) proposed observation-based feedback linearization control; and (j) adaptive-fuzzy technique. The proposed observation-based feedback linearization control has two main subparts: (1) the classical feedback linearization control; and (2) the proposed observation algorithm designed in the previous section and based on the fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization observer. Based on Figures 5 and 6, four PC-based controllers are designed to control four motors located in the slave manipulator. These motors are used to actuate the translation, rotation, deflection, and gripper of the surgical robot. The conventional feedback linearization fault-tolerant control is represented in the following Equation:  
\[\left\{ \begin{array}{l}\tau_{FLC}(k)=-I(q)\times(\ddot{X}_{d}+\alpha_{P}e(k) +\alpha_{d}\dot{e}(k)+\alpha_{i}\dot{\sum}e(k))+f(\hat{X}_{q,F_{x}(p)},U)\\ e(k)=(Y_{q}(k)-\hat{Y}_{q(p)}(k))\end{array} \right., \tag{33}\]  
Here, \(\tau_{FLC}\) and \((\alpha_{p},\alpha_{d},\alpha_{i})\) are the controller output (torque) based on the conventional feedback linearization control algorithm and controller coefficients, respectively. In addition to several positive points of classical feedback linearization control algorithm, to improve the robustness and increase the power of fault reduction, the proposed fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization observer is presented. The main advantage of this part is to estimate the unmeasured variable \((X_{q,F_{x}\tau})\) by observation. The proposed observer feedback linearization control is represented by the following Equation:  
\[\left\{ \begin{array}{l}\tau_{OTLC}(k)=\tau_{FLC}(k)+\hat{Y}_{q(p)}(k)\\ \hat{Y}_{q(p)}(k+1)=(\kappa)^{T}\hat{X}_{q,F_{x}(p)}(k)\\ \hat{X}_{q,F_{x}\tau(p)}(k+1)=[\alpha\chi\hat{X}_{q,F_{x}(p)}+\alpha_{U}U_{F_{ x}\tau}(k)+\alpha_{Y}\hat{Y}_{q(p)}(k)+\alpha_{F}\hat{f}_{fuzzy(p)}]\\ +\hat{\delta}_{(p)}(k)+\alpha_{P}(Y_{q}(k)-\hat{Y}_{q(p)}(k))+f(\hat{X}_{q,F_{ x}(p)},U)+I^{-1}(\alpha_{P}(Y_{q}(k)-\hat{Y}_{q(p)}(k)))\end{array} \right. \tag{34}\]  
Here, \(\tau_{OTLC}(k)\) is the controller output based on the proposed observation-based feedback linearization controller. To improve the reliability of the proposed observation-based feedback linearization fault-tolerant controller, online tuning based on the fuzzy algorithm is proposed. To improve the power of fault reduction in the proposed fault-tolerant control algorithm, the controller coefficients play important roles. Because faults are unknown with respect to various conditions and are difficult to predict, calculating the optimal values for coefficients are complicated. If the controller's coefficients are changed with the condition of the surgical robot for the sinus, the fault could be reduced and eliminated. Thus, for online tuning of the coefficient, a fuzzy algorithm is represented. Based on  
Figure 12: Actuator fault identification residual signals: (**a**) position and (**b**) torque for the fuzzy ARX-Laguerre feedback linearization method, the fuzzy ARX-Laguerre robust feedback linearization method, and the fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization method for the surgical robot for the sinus. Equation (34), the adaptive fuzzy observation-based feedback linearization fault-tolerant controller can be written as the following Equation:  
\[\left\{