# Review of Fault-Tolerant Control Systems Used in Robotic Manipulators
## 3 Sliding-Mode Control in FTCS
The solution proposed in [40] is described by the following Formulas (17)-(20):  
\[u_{stw}=\mu_{1}|e|^{\frac{1}{2}}\text{sign}(e)+\xi, \tag{17}\]  
where \(u_{stw}\) is an approximate unknown component, \(\mu_{1}>0\), \(e=x_{2}-v\) is a velocity error, \(\xi=k(t)\text{sign}(e)\),  
\[\dot{v}=\wedge u+f(x_{1},x_{2})+u_{stw}+k_{1}[e]^{\gamma_{1}}+k_{2}[e]^{\gamma_ {2}}, \tag{18}\]  
where \(\dot{v}\) is the observer that was designed based on a super-twisting high-order sliding-mode algorithm, \(\wedge=M^{-1}(q)\), \(M\) is the inertia matrix, \(k_{1}>0\), \(k_{2}>0\), \([e]^{\gamma_{1}}=[e]^{\gamma_{1}}\text{sign}(e)\), \(v\) is the estimate of the velocity \(x_{2}\), \(\gamma_{1}=\frac{1}{2}\), \(\gamma_{2}>2\),  
\[s=e+\frac{1}{k_{2}^{7}}\big{[}\dot{e}+k_{1}[e]^{\alpha}\big{]}^{\frac{1}{7}}, \tag{19}\]  
where \(s\) is the proposed sliding surface, \(\alpha>1\), \(\frac{1}{2}<\gamma<1\),  
\[u=u_{0}+u_{c}+u_{s}, \tag{20}\]  
where \(u\) is the proposed controller output, \(u_{0}=\wedge^{+}(-f(x_{1},x_{2})-u_{stw}+\ddot{x}_{d}-k_{1}|e|^{\alpha-1}\dot{ e})\), \(\wedge^{+}\) is the pseudo-inverse of the \(\wedge\), \(u_{c}=-\wedge^{+}k_{2}^{7}\big{|}T(e,\dot{e})|^{1-\frac{1}{7}}\dot{e}\), \(u_{s}=\wedge^{+}(-(\Gamma+\alpha)sign(s)-\lambda_{1}[s]^{m_{1}}-\lambda_{2}[s] ^{m_{2}})\), \(\Gamma\) is a positive constant, \(0<m_{1}<1\), \(m_{2}>1\). The proposed algorithm was designed to follow a preset trajectory. The block for generating the preset trajectory is thus the first block of control. The signal from this block, together with the signal feedback from the FxTSOSMO block described by Equations (17) and (18), goes to the summing node. The feedback signal is the current positions and velocities. The position and velocity errors go from the summation node to the sliding surface block described by Equation (19). The signal \(s\) from the sliding surface block and \(u_{stw}\) from the FxTSOSMO block go to the FxTSMC block described by the equation. The output of the FxTSMC block \(u\) described by Equation (20) goes to both the FxTSOSMO block and the robot. Finally, the FxTSOSMO block retrieves information about the robot in the form of positions and velocities. In [42; 43], the authors presented a fault-tolerant control system for a single-link flexible joint manipulator (SFJM). This work proved that it is possible to control the flexible coupling in a robot joint using SMC with SMTO. The presented results show that SMC is a method that can be applied to objects characterized by non-linearity and uncertainty. However, the presented solution was validated only for a single-axis manipulator, but it is possible to develop a similar system for a manipulator with more degrees of freedom. The authors of [44] proposed a novel methodology for manipulator adaptive backstepping non-singular fast terminal sliding-mode control (ABNFTSMC). The proposed approach combined NFTSMC with a backstepping design mechanism. The combination resulted in few tracking errors and low \"chattering\" on the output and provided fast response transients. The described system was compared to CTC, PID, PID-SMC, and NFTSMC controllers. The comparison of these systems showed improvements in the above-mentioned parameters. In [45], a system consisting of combined non-singular fast terminal sliding-mode control (NFTSMC) and a high-order sliding-mode (HOSM) controller was proposed. The presented control system also used an algorithm based on time delay estimation (TDE) for fault estimation. The results described in the work showed that fast terminal sliding-mode control (FTSMC) and NFTSMC systems have faster convergence compared to the non-singular terminal sliding-mode control method. FTSMC systems may encounter the problem of singularity during operation. However, the work shows the advantage of the NFTSMC system over the above-mentioned ones. Other methods using both active fault-tolerant control systems (AFTCS) and passive fault-tolerant control systems (PFTCS) were also rated in the paper.