# Review of Fault-Tolerant Control Systems Used in Robotic Manipulators
## 2 Artificial Intelligence Methods in FTCS  
Artificial intelligence methods are highly developed techniques that have applications in fault-tolerant strategies for controlling robotic arms. Among the artificial intelligence methods used for fault-tolerant control systems are fuzzy logic controllers, which are able to effectively deal with non-linear systems using membership functions, which assess analog input signals using logical variables that take values between 0 and 1. The authors of [17] described the so-called adaptive fuzzy type-2 backstepping control (AFT2 BC) method. The backstepping method was also described in [18]. AFT2 BC was tested on a model of a PUMA560 robotic arm in a MATLAB environment in which the kinematic configuration of the robot could be changed simultaneously to simulate axis failure and changes in the load on the arm. The proposed control algorithm did not require a priori knowledge of the dynamic robot model so the controller could operate when the faulty condition was due to both model uncertainty and external disturbances. Figure 1 shows a block scheme of the proposed adaptive control method. On the description of the graph is a set of Lyapunov Equations (1)-(6) [17]:  
\[\dot{V}_{1}=e_{1}\dot{e}_{1}=e_{1}(\dot{q}_{d}-x_{2}), \tag{1}\]  
where \(\dot{V}_{1}\) is the derivative of the Lyapunov function [19], \(e_{1}\) is the tracking error, \(\dot{q}_{d}\) is the derivative of the desired signal, \(x_{2}\) is the vector of the current state \(x_{2}=\dot{x}_{1}\), \(x_{1}=\big{[}q_{1}q_{2}q_{3}\big{]}^{T}\),  
\[u_{a}=W^{T}(e_{1}\dot{e}_{1})\Theta, \tag{2}\]  
where \(u_{a}\) is the fuzzy type-2 adaptive control expression that is designed to estimate the ideal backstepping control law, \(W^{T}\) represents the average basis functions obtained by a fuzzy type-2 system, where each basis function is given by the average of the corresponding left and right basis functions, \(\Theta\) denotes the adapted vector parameters,  
\[\dot{\Theta}=\gamma e_{2}W\big{(}e_{1}\dot{e}_{1}\big{)}-\sigma_{1}\Theta, \tag{3}\]  
where \(\gamma>0\), \(\sigma_{1}>0\), \(e_{2}=\dot{q}_{d}+c_{1}e_{1}-x_{2}\), \(c_{1}\) is a positive constant vector, \(\Theta(0)=0\),  
\[u_{r}=\hat{\epsilon}\text{tanh}\bigg{(}\frac{e_{2}}{\chi}\bigg{)}, \tag{4}\]  
where \(u_{r}\) is the robust control expression that is introduced to reduce the effects of fuzzy type-2 estimation errors,  
\[\dot{\hat{\epsilon}}=\eta\hat{\epsilon}_{2}\text{tanh}\bigg{(}\frac{e_{2}}{ \chi}\bigg{)}-\sigma_{2}\hat{\epsilon}, \tag{5}\]where \(\eta>0,\sigma_{2}>0,\chi>0,\ell(0)=0\) are parameters,  
\[u_{p}=c_{2}e_{2}, \tag{6}\]  
where \(c_{2}>0\). The authors of [20] presented a fault-tolerant control scheme in the form of non-singular terminal synergetic control (NTSC) combined with interval type-2 fuzzy satin bowerbird optimization (IT2 FSBO). The paper also described an adaptive augmented extended Kalman filter (A-AEKF) that detected, identified, and isolated actuator faults, even in the presence of noise [21]. The authors of [22] presented a neuro-fuzzy (NF) robot fault-detection algorithm that allowed for the control of a robot with an SRI [23] controller despite sensor or axis actuator failures. The detection and fault-tolerance architecture of the proposed solution was built with a multilayer perceptron trained with the backpropagation of errors and a fuzzy logic block. The input layer of the artificial neural network consisted of 15 neurons organized into 3 groups, each consisting of 5 neurons, 1 for each axis. The three groups were used in turn for the preliminary assessment of the position, velocity, and acceleration. The network had two groups of output neurons, each with five neurons, one for each axis. These groups generated the positions and velocities for each axis in turn. The results of the multilayer perceptron (MLP) in the form of the product of the position and velocity, i.e., \(q\dot{q}\), was summed with the output parameters of the robot. This sum was sent to the inference block, along with the results of the fuzzy logic block.