# Review of Fault-Tolerant Control Systems Used in Robotic Manipulators
## 4 Other Robot-Dedicated FTCS  
In addition to the leading fault-tolerant manipulator control methods such as SMC and artificial intelligence, other dedicated manipulator control methods can also be found in the literature. Some of them are presented in this section of the article. Figure 8: Structure of the NFTSMC-TOSMO control method [49]. The authors of [50] presented the use of the well-known Newton-Raphson Method (NRM) [51] to control a manipulator when a robot axis fails. The authors of this paper attached a QR code to the robot's end arm. This code was read by a camera located at the base of the robot and then the position of the robot's end was fed into the control software. This allowed the algorithm to determine the moment at which the failure occurred by not matching the position tracked by the camera with the calculated position. Then, the system, which used the Newton-Raphson method in the inverse kinematics calculation to find a good approximation for the root of a real-valued function, determined which joint had failed. To determine the faulty axis, the position measured by the camera was compared with the position calculated by the NRM. If a difference was detected, the system checked which axis was damaged. If a faulty joint was detected, the failed joint was treated as rigid and the robot's inverse kinematics were recalculated using the Newton-Raphson method. Using this solution, the authors of the paper proved that if the robot was able to reach the set position despite the loss of an axis, the algorithm was able to help the robot achieve its goal. In [52], a new method of hybrid fault-tolerant predictive control (HFTPC) was proposed. This method was tested on a hybrid manipulator that was a combination of hydraulic and mechanical elements. The paper compared the proposed HFTPC method, one of the components of which was a predictive control (MPC) model [53], with control in which only the MPC part was running. The presented system was able to control a hybrid manipulator when a failure occurred. The authors of [54] presented robust fault-tolerant tracking control (RFTTC) schemes for uncertain non-linear feedback systems (NFS) using operator theory-based robust right coprime factorization (ORRCF). It was shown that both RFTTC based on an internal model and RFTTC based on operator compensation were effective in dealing with interference and erroneous signals. The proposed system was tested on a trajectory-tracking task by a two-axis robot. The authors of [55] presented an adaptive fault-tolerant control system for unknown actuator faults. Detailed fault information was not required for the algorithm to work. The proposed method omitted complex logarithmic transformations in the pre-scribed performance control (PPC) structure [56], making the controller efficient and simple to implement. The authors of [57] proposed a robust fixed-time fault-tolerant control system (RFTTC) by introducing an additional auxiliary vector that was described by Formula (26) [57]:  
\[\chi=\dot{e}+\lambda e, \tag{26}\]  
where \(\lambda\in R^{+}\), \(e=q-q_{d}\) is the position error and \(\dot{e}=\dot{q}-\dot{q}_{d}\) is the velocity error. A schematic of the proposed system is shown in Figure 9.  
The authors of [58,59] presented an FTCS for manipulators based on an active inference controller (AIC). AIC makes use of the sensory prediction error in the free energy for the generation of residuals and thresholds for FDI. It does not require additional controllers for fault recovery. In [59], the AIC was improved, and unbiased AIC (u-AIC) was introduced to reduce the probability of false positives and allow for the easy definition of probabilistically robust thresholds for fault detection. Figure 9: RFTC with auxiliary vector [57]. The authors of [43] presented fault-tolerant control using a proportional-integral-derivative (PID) controller. In addition to the PID controller, the system consisted of a Luenberger observer [60], which was able to estimate the errors of the FDI analysis.