# Review of Fault-Tolerant Control Systems Used in Robotic Manipulators
## 3 Sliding-Mode Control in FTCS
In [46], a combination of a controller in the form of non-singular fast terminal sliding-mode control (NFTSMC) with an observer in the form of third-order sliding mode (TOSM) was presented. The research results showed that the TOSM observer can estimate the speed of the system so that the system does not need to measure this. This method was compared with the SMC, NTSMC, and NFTSMC methods. The proposed NFTSMC method had the fewest trajectory-tracking errors. The TOSM observer, on the other hand, proved more accurate and had less \"chattering\" than the SOSM observer. The authors of [47] presented a combination of a fast terminal sliding-mode surface (FTSMS), super-twisted reaching control law (STRCL), and disturbance observer (DO). The proposed method was compared with the SMC and NFTSMC methods. The method described in the paper provided convergence in finite time and effectively combatted the chattering phenomenon. By using a disturbance observer, the complexity of the calculations was reduced. A DO was also used to estimate uncertainties in the form of dynamics, external disturbances, and failures. A diagram of the proposed method is shown in Figure 7. This work presents the new fast terminal sliding surface described by the formula in [47]:  
\[\sigma_{i}=\hat{e}_{i}+\frac{2\gamma_{1}}{1+E^{\mu_{1}(|\hat{e}_{i}|-\Phi)}}e_{ i}+\frac{2\gamma_{2}}{1+E^{\mu_{2}(|\hat{e}_{i}|-\Phi)}}|e_{i}|^{\alpha}\text{ sign}(e_{i}), \tag{21}\]  
where \(\sigma\in R\) is the FTSMS, \(E\) is an exponential function, \(e_{i}\) represents the positional control error, \(\hat{e}_{i}\) represents the velocity control error, \(\gamma_{1},\gamma_{2},\mu_{1},\mu_{2}\) are the positive constants, \(0<\alpha<1\), \(\Phi=\left(\frac{\gamma_{1}}{\gamma_{2}}\right)^{1/(1-\alpha)}\),  
Figure 5:0  
where \(u_{r}\) is STRCL, \(Y_{1}=\text{diag}(Y_{11},\dots,Y_{1n})\). The authors of [48] proposed a new sliding-mode control technique, namely active fault-tolerant control with synchronous sliding-mode control AFTC-SSMC. This control relied on the fact that with a traditional SMC controller, only the position error converges to zero. In synchronous control, on the other hand, both the position error and kinematic relationship between the errors converge to zero. The method was compared with standard active and passive sliding-mode control, namely AFTC-SMC and PFTC-SMC. Figure 5:1 Diagram FTSMS-STRCL-DO [47]. It was shown that the pro-rated method had better fault tolerance and provided better trajectory-tracking performance. The authors of [49] presented a fault-tolerant manipulator control strategy in the form of a combination of NFTSMC and the proposed high-speed third-order sliding-mode observer TOSMO. This combination was able to handle unknown input data, further reducing the vibration phenomenon and improving the accuracy of trajectory tracking. The proposed high-speed TOSMO could estimate both the velocity signal and input data faster than a standard TOSMO. The structure of such a system is similar to that of the FTSMS-STRCL-DO [47] but it has a different observer model. The structure of the proposed method is shown in Figure 5:2 The solution proposed by the authors is described by the following formulas [49]:  
Figure 5:3  
where \(\hat{s}\) is the NFTS surface, \(\beta_{1},\beta_{2},\beta_{3},\beta_{4}\) are positive constants, \(0<\gamma_{1}<1\), \(\gamma_{2}=2\gamma_{1}/(1+\gamma_{1})\) are constants,  
Figure 5:4  
where \(u_{eq}\) is the equivalent control law, \(A=\beta_{1}|e|^{\gamma_{1}}sign(e)+\beta_{2}\hat{\hat{e}}^{\gamma_{2}}sign \Big{(}\hat{\hat{e}}\Big{)}+\beta_{3}e+\beta_{4}e^{3}\), \(\psi(x)=M(q)^{-1}\big{[}-C\big{(}q,\dot{q}\big{)}-G(q)\big{]}\) is the nominal model of the robot manipulator, \(\Gamma\) is a positive constant, \(k_{2},k_{3}\) denotes the sliding gains, \(\tilde{x}_{d}\) is an expected acceleration  
Figure 5:5  
where \(u_{sw}\) is the switching control law, \(\mu\) is a small positive constant, and \(\Delta_{\delta}\) is the estimation error as a positive value.