## 4 Results and Analysis
### Fault Detection and Fault Identification  
To analyze the effectiveness of the proposed algorithm for fault detection and identification, normal and abnormal operating conditions (actuator fault, sensor fault, and actuator-sensor fault) are considered. Figure 2 illustrates the torque and position of the robot manipulator's joint in normal and faulty actuator conditions. The actuator fault, sensor fault and actuator-sensor fault for the PUMA robot manipulator are defined by Equations (59)-(61), respectively. \[Actuator\;Fault=\left\{ \begin{array}{c}\Theta_{a}=|0.2|\;(N.m)\\ \Theta_{s}=0\end{array} \right. \tag{59}\]  
\[Sensor\;Fault=\left\{ \begin{array}{c}\Theta_{a}=0\\ \Theta_{s}=|0.015|\;(Deg)\end{array} \right. \tag{60}\]  
\[Actuator-Sensor\;Fault=\left\{ \begin{array}{c}\Theta_{a}=|0.2|\;(N.m)\\ \Theta_{s}=|0.015|\;(Deg)\end{array} \right. \tag{61}\]  
where \(\Theta_{a}\) and \(\Theta_{s}\) are actuator fault and sensor fault values, respectively. To test the accuracy, robustness and repeatibility of the proposed algorithm, three different faults were seeded on the desired signal at different locations and absolute value: actuator, sensor and actuator-sensor. To test the robustness of the proposed ANHVSO, NHVSO and VSO methods under noisy conditions, Gaussian noise was added to the original signals in the normal and faulty conditions. Figures 3-5 illustrate the position and torque joint residual signals in normal and faulty conditions based on the VSO, NHVSO and ANHVSO for the robot manipulator, respectively. Regarding Figure 3, the VSO is prone to high frequency oscillation, especially under faulty conditions. To improve the robustness and accuracy, and to reduce the high frequency oscillation, NHVSO is used in the current work. Figure 4 illustrates the torque and position residual signals in the presence of noise for the robot manipulator under normal and faulty conditions. According to the residual signals computed through Equation (22) and illustrated in Figures 3 and 4, we can see that the VSO and NHVSO have problems to effectively estimate the signals under normal and abnormal conditions. Specifically, it means that the residual signals corresponding to some system states that were derived by the aforementioned methods have severe overlap (i.e., they are not discriminative enough). This problem can affect the final fault classification accuracy and increase the misclassification rates. From these figures, it can be observed that the residual signals are well differentiable in normal conditions, which means that they can be used for fault detection (i.e., differentiation between normal and abnormal conditions); but these techniques are not suitable for classification of the faults. Figure 5 illustrates the position and torques residual signals under the proposed ANHVSO algorithm, considering the normal and abnormal conditions of the PUMA robot manipulator. ```
1 Robot manipulator modeling based on the Lagrange method (6). 2 Run the second order VSO (8,9). 3 Improve the performance of the second order based on the HVSO (11,12). 4 Increase the reliability of this HVSO based on the neural algorithm and design the NHVSO (16,17). 5 Increase the fault estimation reliability and accuracy in the NHVSO with the ANHVSO (18,20,21). 6 Run the residual signal characterization by energy (22,23). 7 Run the learning process for SVM and apply the SVM classification technique for fault detection and identification (24-40). 8 Run the robust FTC based on the VSC (41). 9 Increase the robustness of the VSC based on a backstepping technique to design the BVSC (42). 10 Increase the accuracy of the BVSC in uncertain conditions using a fuzzy algorithm and design
11 FBVSC (55). 12 Reduce the effects of faults and increase the reliability of the FBVSC based on the proposed
13 observation technique and the designed MFBVSC (56). 14 Increase the robustness, stability, accuracy and performance, and reduce chattering in the MFBVSC using the adaptive technique and the designed AMFBVSC (57,58).