## 1 Introduction
Both use a synchronization technique based on the synchronous terminal sliding mode control (S-TSMC) and the extended state observer. First, to estimate lumped uncertainties, disturbances, and faults, an extended state observer (ESO) [22] is adopted. An ESO is a simple technique for estimating faults in which simply adjusting the observer parameters leads to simple application in real systems. Next, an AFTC using the synchronization technique based on terminal sliding mode control (TSMC) [23] with a novel synchronization error and coupling position error (AFTC S-TSMC1) is proposed. The novel synchronization error can approach zero in a finite time compared to the conventional coupling position error in [24]. In addition, the novel synchronization error has an advantage over the conventional error [25], because it is more closely related to other joints. Compared with existing finite-time synchronization controls, such as [26], the novel synchronization error does not lead to a singularity when the desired trajectory crosses zero. This improves the range of the robot manipulator over that in [26]. Some authors (such as in [27; 28]) use graphic theory and prescribe performance control in the synchronization control. However, the conventional prescribed performance term can become a singularity during operation, and graphic theory may not be suitable for a single robot manipulator controller. The second proposed AFTC based on the synchronization technique (AFTC S-TSMC2), but without the coupling position error, is designed to improve the synchronization of the synchronous terminal sliding mode control. With the combination of ESO and the synchronization technique, the two proposed AFTC S-TSMCs can avoid the drawbacks of the PFTC strategy, and their fast response leads to a robot system that can deal with high-magnitude faults while reducing the picking phenomenon. Finally, both simulated and experimental results from the two proposed AFTCs verify the effectiveness of the two novel synchronous terminal sliding mode controllers. The contributions of this paper are summarized as follows:  
(1)Two active fault-tolerant control algorithms for robot manipulators, based on novel finite-time synchronous terminal sliding mode controllers and an extended state observer, are proposed. The novel finite-time synchronization technique has the ability to make both the joint position error and the synchronization error simultaneously approach to zero. Due to these internal constraints of the synchronization control, the proposed controller can make the system quickly respond to the faultsin a forward way before its feedback response after a fault estimation. Therefore, the proposed controller can reduce the occurrence of the picking phenomenon due to the slow feedback response of AFTC strategy. (2) The novel synchronization error leads to better synchronization because it uses more information from other joints than the conventional synchronization error in [25], which has the information from only one neighbor joint. (3) The novel coupling error can make the position error approach zero in a finite-time while the conventional coupling error in [24] cannot guarantee the finite-time convergence. (4) Two proposed AFTCs can avoid the singularity in both the desired trajectory and the control action, while the control algorithms in [26; 27; 28] cannot guarantee avoiding the singularity. This ability allows increase in the workspace of the robot. (5) Experimental results show the effectiveness of the proposed AFTC in reducing both the picking phenomenon and in handling faults of high magnitude. The rest of this paper is organized as follows. In Section 2, the dynamic model of a robot manipulator and associated faults are presented. Fault estimation based on the extended state observer is discussed in Section 3. In Section 4, a novel synchronization error, coupling position error, and the finite-time active fault-tolerant control using a synchronous fast terminal sliding mode control are proposed.