## 1 Introduction  
In robot manipulator systems, the occurrence of faults or failures in the actuators or sensors may lead to degraded robot system performance, system breakdown, and economic loss. In response to the requirements of enhanced reliability and safety, fault-tolerant control (FTC) has attracted the attention of researchers over the past few decades. FTC strategies can be divided into two main types [1]: passive FTC (PFTC) [2; 3], and active FTC (AFTC) [4; 5; 6]. In PFTC, the ability to tolerate abnormal operation in the presence of faults/failures in components mainly depends on the robustness of the controller, which can use sliding mode control (SMC) [7; 8] or adaptive control [9; 10]. For example, in SMC, faults are considered to be external disturbances. To guarantee the stability of a system, knowledge of the bounded values of the faults is required in the SMC design. However, when the bounded value of a fault has a high magnitude, oscillation can occur during normal operation of a system due to the problem of high gain control. Therefore, the ability of PFTC to deal with high magnitude faults is limited. Unlike PFTC, AFTC uses the fault information from the fault estimation process to compensate the conventional controller. Fault estimation (FE) is a powerful technique that includes fault detection, fault isolation, and fault identification within a step. The accuracy of the FE highly affects the performance of AFTC, so many estimation techniques have been developed to improve FE accuracy, such as sliding mode fault estimation [11; 12], the extended state observer technique [6; 13], and the learning observer technique [4]. The combination of FE and a conventional controller not only helps AFTC handle faults with a high magnitude, but also overcomes the high gain control issue associated with PFTC. Most work on developing AFTC focuses on improving the FE accuracy [14; 15] or increasing performance of the controller [5; 16]. Those AFTC approaches have led to acceptable performance, but the slow response issue still exists, causing the picking phenomenon after a fault occurs. This strongly affects the performance of AFTC. Therefore, in this paper, using the concept of synchronization control, reduction of the picking phenomenon is addressed to increase the performance of AFTC. Synchronization control is known as an effective controller for close-loop chain mechanism systems, such as parallel robot manipulators [17], cable-driver parallel manipulators [18; 19], dual-drive gantry mechanisms [20], and cooperation robot manipulators [21]. Due to the constraints of the close-loop chain and the existence of position errors at the actuators during motion of the system, tensor internal forces may occur. This type of force is considered to be an internal uncertainty that can degrade the performance of a conventional controller. Using a synchronization controller, the position errors can simultaneously be equal and tend to zero, reducing the effect of the internal force. Therefore, the synchronization control technique can improve performance of a close-loop chain system. In an open-chain system, such as a serial robot manipulator, this kind of internal force may not exist, so the synchronization technique is not effective. However, the position errors at each joint still simultaneously approach zero when using synchronization control. In this paper, we use the synchronization technique in a fault-tolerant controller to reduce the effect of the picking phenomenon associated with the AFTC strategy (AFTCs). When a fault occurs at an actuator, a controller using the synchronization technique can make the position error at each joint equal, so the controller can quickly respond to a fault before the controller has the feedback information from the fault estimation process. Therefore, the picking phenomenon can be reduced. In this paper, two finite-time active fault-tolerant controls for robot manipulators are proposed.