# 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 Detection, Estimation, and Identification Technique  
As identified in Figure 2, the fault detection, estimation, and identification blocks have two sub-blocks: (g) residual and threshold generator and (h) residual bank and logic decision. The residual signal is generated by Equation (29).  
\[r=Y_{(q)}-\hat{Y}_{q(p)} \tag{29}\]  
Figure 10: Results for the fuzzy ARX-Laguerre feedback linearization method, fuzzy ARX-Laguerre robust feedback linearization method, and fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization method for the surgical robot for the sinus. (**a**) Position fault estimation accuracy. (**b**) Fault estimation error of position estimation.  
Based on previous work [18; 23; 26; 27], the robust sliding mode algorithm is used to generate the threshold value. The sliding mode threshold value is generated by Equation (30).  
\[\left\{\begin{array}{l}\text{{TH}}_{N}=\alpha_{N}[\text{sgn}(\lambda_{N} \times r_{N}+(\frac{\lambda_{N}}{2})^{2}\sum r_{N}+\dot{r}_{N})]\\ \text{{TH}}_{a}=\alpha_{a}[\text{sgn}(\lambda_{a}\times r_{a}+(\frac{\lambda_ {a}}{2})^{2}\sum r_{a}+\dot{r}_{a})]\\ \text{{TH}}_{s}=\alpha_{s}[\text{sgn}(\lambda_{s}\times r_{s}+(\frac{\lambda_ {a}}{2})^{2}\sum r_{s}+\dot{r}_{s})]\end{array}\right.. \tag{30}\]  
Here, \(\text{{TH}}_{N},\text{{TH}}_{a},\text{{TH}}_{s},(\alpha_{N},\alpha_{a},\alpha_ {s},\lambda_{N},\lambda_{a},\lambda_{s}),r_{N},\dot{r}_{N},\sum r_{N}r_{a}, \dot{r}_{a},\sum r_{a},r_{s},\dot{r}_{s},\) and \(\sum r_{s}\) are the threshold value for normal condition, threshold value for actuator fault, threshold value for sensor fault, coefficients of the sliding mode algorithm in different states, residual signal for normal condition, change of residual signal for normal condition, integral of residual signal for actuator fault, change of residual signal for actuator fault, integral of residual signal for sensor fault, and integral of residual signal for sensor fault, respectively. Two different conditions have been defined for fault detection in the surgical robot for the sinus: (a) normal condition \((\delta(k)=0)\) and (b) abnormal condition \((\delta(k)\
eq 0)\). The fault detection in the manipulator of the surgical robot for the sinus is represented by Equation (31).  
\[\left\{\begin{array}{l}\text{Normal Condition}:r(k)<\text{{TH}}_{N}\\ \text{abnormal Condition}:r(k)\geq\text{{TH}}_{N}\end{array}\right.. \tag{31}\]  
Figure 11 illustrates the fault detection for the fuzzy ARX-Laguerre feedback linearization observer, fuzzy ARX-Laguerre robust feedback linearization observer, and proposed algorithm. Based on the fuzzy ARX-Laguerre fuzzy robust feedback linearization observer, the proposed high-accuracy fault estimation technique is represented by Equation (27). Based on Equations (26)-(31), the following equation is used for fault identification for the manipulator of the surgical robot for the sinus.  
\[\left\{\begin{array}{l}(\text{Actuator})\text{ Fault}:(r_{a}(k)\geq\text{{TH}}_{a}\text{ {\&}}\text{{TH}}_{N}\leq r_{s}(k)<\text{{TH}}_{s})\\ (\text{Sensor})\text{ Fault}:(\text{{TH}}_{N}\leq r_{a}(k)<\text{{TH}}_{a}\text{ {\&}}\text{{r}}_{s}(k)\geq\text{{TH}}_{s})\\ (\text{Sensor}\text{ }-\text{ Actuator})\text{ Fault}:(r_{a}(k)\geq\text{{TH}}_{a}\text{ {\&}}\text{{r}}_{s}(k)\geq\text{{TH}}_{s})\end{array}\right.. \tag{32}\]  
Figure 12 illustrates the fault identification in the manipulator of the surgical robot for the sinus for the fuzzy ARX-Laguerre feedback linearization observer, fuzzy ARX-Laguerre robust feedback linearization observer, and the proposed algorithm. Based on fault estimation accuracy, the proposed algorithm has better performance for fault diagnosis time delay.  
Figure 11: Fault detection 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.