# Adaptive Fuzzy-Based Fault-Tolerant Control of a Continuum Robotic System for Maximly Sinus Surgery
## 4 Results and Analysis
### Fault-Tolerant Control  
The root mean square (RMS) error of the kinematics and the end-effector position for the surgical robot for the sinus in the normal, actuator fault, sensor failure, and actuator-sensor fault conditions are illustrated in Figures 18-21. Based on Figure 18, the RMS error in the normal condition for the end-effector of the surgical robot for the sinus is close to zero. In Figures 19-21, the error rate is greater than at the normal condition. Regarding these figures, the end-effector positions fluctuate similar to in the normal condition. After fault detection, estimation, and identification based on the fuzzy ARX-Laguerre T-S fuzzy robust feedback linearization (proposed) observer, the proposed fault-tolerant control based on the adaptive fuzzy proposed observation-based feedback linearization controller is tested. To test the proposed fault-tolerant controller, the proposed method is compared with the observation-based feedback linearization controller and the classical feedback linearization control algorithm. The main goal in this part is elimination or reduction of the effect of the fault to increase the rate of successful surgery. Figure 22 shows the effects of the proposed algorithm, the observation-based feedback linearization algorithm, and the classical feedback linearization fault-tolerant controller to reduce/eliminate faults. Based on this figure, the adaptive fuzzy observation-based feedback linearization controller is more reliable for the fault-tolerant controller than the other two methods. The effect of the proposed fault-tolerant algorithm on the RMS error of kinematics end-effector and position is illustrated in Figure 23. Regarding this figure, the proposed technique sharply reduces the RMS error and fluctuation of the end-effector. According to Figures 18 and 23, it is clear that the adaptive fuzzy observation-based feedback linearization control is an active and accurate technique for fault-tolerant control in the surgical robot for the sinus. Figure 17: (**a**) Position residual signals and (**b**) torque residual signals for **actuator-sensor** fault identification based on 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. Figure 19: Actuator fault condition: (**a**) kinematics end-effector position RMS error and (**b**) end effector position in the XYZ plot. Figure 20: Sensor fault condition: (**a**) kinematics end-effector position RMS error and (**b**) end effector position in the XYZ plot. Figure 18: Normal condition: (**a**) kinematics end-effector position root mean square (RMS) error and (**b**) end effector position in the XYZ axis. Figure 24 illustrates the fault-tolerant accuracy of the proposed fault-tolerant controller and feedback linearization fault-tolerant controller in normal and abnormal conditions. This figure illustrates that the minimum error, the average error, and the maximum error for the normal condition with the proposed fault-tolerant controller are 1.8, 2.3, and 3.6 mm, respectively. For the faulty condition using the proposed fault-tolerant algorithm, the errors are 0.8, 2.45, and 7.1 mm, respectively. For the faulty condition using the observer feedback linearization fault-tolerant controller are 6, 8.4, and 20 mm, respectively, and those for the faulty condition with respect to the classical feedback linearization  
Figure 21: Actuator-sensor fault condition: (**a**) kinematics end-effector position RMS error and (**b**) end effector position in the XYZ plot. Figure 22: (**a**) Fault-tolerant control error and (**b**) error of position fault for the fuzzy feedback linearization method, observation-based feedback linearization method, and proposed method for the surgical robot for the sinus. Figure 23: Power of proposed fault-tolerant algorithm for (**a**) kinematics end-effector position RMS error and (**b**) end effector position in the XYZ plot.