# 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
### Takagi-Sugeno (T-S) Fuzzy Advanced Observer  
Following the abbreviations shown in Figure 2, the fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization observer designed in this section comprises three main blocks: (d) the fuzzy ARX-Laguerre feedback linearization observer, (e) improvement of the robustness and estimation performance based on the fuzzy ARX-Laguerre robust (variable structure) feedback linearization observer algorithm, and (f) reduction of the estimation error of the fuzzy ARX-Laguerre robust feedback linearization observer based on the T-S fuzzy algorithm and formulation of the fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization technique. The feedback linearization observer offers a non-linear model-reference approach to find an accurate system and fault estimation. Based on Equation (16), the fuzzy ARX-Laguerre feedback linearization observer is proposed for a surgical robot for the sinus based on the following Equation (26):  
\[\left\{ \begin{array}{l}\hat{X}_{q,F_{x}\tau(FLO)}(k+1)=[\alpha_{X}\hat{X}_{ q,F_{x}\tau(FLO)}+\alpha_{U}U_{F_{x}\tau}(k)+\alpha_{Y}\hat{Y}_{q(FLO)}(k)+\alpha_{F}f_{fuzzy(FLO)}]\\ +\hat{\delta}_{(FLO)}(k)+\alpha_{P}(Y_{q}(k)-\hat{Y}_{q(FLO)}(k))+f(\hat{X}_{ q,F_{x}\tau(FLO)},U)+I^{-1}(\alpha_{P}(Y_{q}(k)-\hat{Y}_{q(FLO)}(k)))\\ \hat{Y}_{q(FLO)}(k+1)=(\kappa)^{T}\hat{X}_{q,F_{x}\tau(FLO)}(k)\end{array} \right.. \tag{17}\]  
where \(f(\hat{X}_{q,F_{x}\tau(FLO)},U)=I^{-1}[F_{c}\tau-N(q,\dot{q})-G(q)]\). Here, \(\hat{X}_{q,F_{x}\tau(FLO)}(k),\hat{Y}_{q(FLO)}(k),\hat{f}_{fuzzy(FLO)},\hat{ \delta}_{(FLO)}(k)\), and \(\alpha_{p}\) are the state estimation, output estimation, fuzzy estimation function, fault estimation, and a coefficient, respectively. Except for \(\alpha_{p}\), all are based on the fuzzy ARX-Laguerre feedback linearization observer. The fault is estimated in the fuzzy ARX-Laguerre feedback linearization observer based on Equation (18). \[\hat{\delta}_{(FLO)}(k+1)=I^{-1}(\hat{\delta}_{(FLO)}(k)+\alpha_{P}(Y_{q}(k)- \hat{Y}_{q(FLO)}(k))) \tag{18}\]  
The state and fault estimation errors in the fuzzy ARX-Laguerre feedback linearization observer are represented by Equation (19). \[\left\{ \begin{array}{l}\widetilde{X}_{q,F_{x}\tau(FLO)}(k)=X_{q,F_{x}\tau} (k)-\hat{X}_{q,F_{x}\tau(FLO)}(k)\\ \widetilde{\delta}_{(FLO)}(k)=\delta(k)-\delta_{(FLO)}(k)\end{array} \right. \tag{19}\]  
Figure 9: Proposed fuzzy ARX-Laguerre method, ARX-Laguerre technique, and ARX algorithm position estimation of the surgical robot for the sinus under abnormal conditions. (**a**) Measured and estimated position. (**b**) Error of estimated position. Here, \(\widetilde{X}_{q,F,\tau(\text{(}FLO)}(k)\) and \(\widetilde{\delta}_{\text{(}FLO)}(k)\) are the state estimation error and fault estimation error, respectively. Both are based on the fuzzy ARX-Laguerre feedback linearization observer. To improve the robustness of the fuzzy ARX-Laguerre feedback linearization observer, the fuzzy ARX-Laguerre extended feedback linearization observer based on the variable structure technique is recommended. Based on previous work [18; 23; 26; 27], the variable structure technique is a robust, stable, and reliable method for fault estimation. Thus, to improve the reliability, robustness, and estimation accuracy, the fuzzy ARX-Laguerre variable structure feedback linearization observer is a respectable candidate. The variable structure fault estimation technique is represented by Equation (20). \[\dot{\delta}_{VSO}(k+1)=(\kappa_{VSO})\times\text{sgn}(Y_{q}(k)-\hat{Y}_{q( \text{(}FLO)}(k)) \tag{20}\]  
Based on Equations (17), (18), and (20), the state-space and fault estimation of the fuzzy ARX-Laguerre robust (variable structure) feedback linearization observer are represented by Equations (21) and (22). \[\left\{