# Review of Fault-Tolerant Control Systems Used in Robotic Manipulators
## 3 Sliding-Mode Control in FTCS  
One of the most popular methods used to control robotic arms is sliding-mode control (SMC). Sliding-mode control is based on variable structure systems consisting of independent structures with different properties and switching logic between them. In SMC, a discontinuous control signal, for example, \(+k\) and \(-k\), are used, which forces the system to \"slide\" along the cross-section of the normal characteristics of the system. When a system moves along a curved or sliding surface, the system is said to be sliding. An up-to-date overview of all the available methods of control using the sliding mode was presented in [34]. The simplest sliding-mode control method can be described by the formula:  
\[u=k\cdot\text{sign}(e), \tag{16}\]  
Figure 5: Robust control system architecture [33]. where \(u\) is the calculated signal, \(k\) is a constant that is the magnitude of change, \(e\) is the control signal error, and sign() is the sign function. Equation (16) describes the simple sliding-mode control method. The advantage of this type of control is its simplicity and insensitivity to system uncertainty. Due to the fact that the method is discontinuous and most control systems are discrete, the main disadvantage of the presented method (16) is the system output signal oscillations called chattering. Nevertheless, the method has become attractive to researchers due to its simplicity; therefore, various sliding-mode control models are currently being designed and implemented for different objects. Due to its robustness to uncertainties, the SMC method is also often used in fault-tolerant control systems. It is applicable to manipulators for which fault tolerance is required. One of the SMC fault-tolerant methods is the use of a sliding-mode observer (SMO) [35] for fault detection. The use of a non-linear observer makes it possible to estimate the immeasurable state and model the uncertainty, which allows the construction of an error estimation algorithm. In [36], a fault-tolerant control system in the form of a super-twisting third-order sliding-mode (STW-TOSM) observer and a super-twisting second-order sliding mode (STW-SOSM) controller was presented. This method makes it possible to estimate errors and uncertainties without measuring the speed of the robot's members. For the correct operation of the system, only position measurements are needed. In the paper, the authors presented a comparison of the following systems: the traditional computed torque controller (CTC) [37], the active CTC-FTC controller [38], the passive SM-FTC [39] controller, and the proposed STW-SOSM-FTC controller in both passive and active forms. The test of the proposed controller showed fewer errors, better stability, and no vibrations on the obtained output waveforms. A block diagram of the arrangement of each block in the control process is presented in Figure 6.  
The authors of [40] presented a system in the form of a fixed-time second-order sliding-mode observer (FxTSOSMO) and a fixed-time sliding-mode controller (FxTSMC). The work compared the proposed observer (FxTSOSMO) with the fixed-time sliding-mode observer (FxSMO) designed in an earlier work. The work compared the computed torque controller, non-singular fast terminal sliding-mode control (NFTSMC) controllers [41], and the proposed method (FxTSMC). The latter system achieved higher estimation precision  
Figure 6: Block diagram of the method presented by [PERSON], [PERSON], and [PERSON] [36]. and the error of the proposed solution converged within a fixed time frame.