## 1 Introduction  
Closed-kinematic chain manipulators (CKCMs) for which its motion is achieved in all degrees-of-freedom (DOF) by the combined motion of their active joints can provide higher positioning accuracy and greater payload handling capability than the conventional open-kinematic chain manipulators (OKCM) composed of serial linkages or rigid bodies [1; 2; 3; 4]. Despite the above advantages, CKCMs possess several drawbacks such as synchronization among actual joints and end-effectors, limited workspace, and nonexistent closed-form solutions of forward kinematics. To address the above issues, the concept of synchronization control has been considered, and as a result, there has been much effort in the development and implementation of error synchronization-based control schemes for CKCMs. In a synchronization-based control scheme, all joints are synchronously driven to improve CKCM's performance. The synchronization concept was first introduced in [5], and it was applied to perform tracking control of parallel robots [6; 7; 8; 9]. Most existing synchronized control schemes are model-based such as the computed torque control (CTC) [10], adaptive control [11; 12], and sliding mode control (SMC) [13; 14]. In general, a SMC scheme consists of a driving component that forces the system's trajectory to reach a stable hyperplane (sliding surface) and a design of a sliding surface that assures the plant's desired error dynamics. The implementation of the above synchronized control schemes requires a precise dynamical model of the manipulator, for which its calculation is highly computationally intensive. Moreover, it was concluded that an accurate mathematical dynamic model of CKCMs is difficult to obtain. Consequently, the above model-based control schemes are not suitable for real-time control of CKCM manipulators, particularly those with more than 2 DOFs. To tackle the above dynamic modeling issue, the authors in [15; 16] considered synchronized control schemes that have simple structures and do not require knowledge of the manipulator dynamics to implement their control laws. However, those control approaches proposed for parallel manipulators can only achieve asymptotic stability, which requires infinite time to converge to an equilibrium point. In order to assure finite-time convergence, the terminal SMC (TSMC) scheme was proposed in [17; 18; 19]. Then, the advantage control scheme of TSMC, called Nonsingular Fast Terminal Sliding Mode Control (NFTSMC), was introduced in [20; 21]. This developed control scheme can handle the singularity and fast convergence of the system. Recently, the NFTSMC was combined with synchronization, and this control scheme was applied to a parallel robot manipulator in [7]. It used the cross-coupling error that combined both tracking errors among the active joints and synchronization errors of a parallel robot to fix the actuator's external disturbances and dynamic uncertainties. Thus, the tracking performance of the robot improved significantly. However, the gains of this control scheme are still selected based on conservative estimates of the dynamic manipulator model. Thus, it leads to complications in the highly complicated model in calculations. Recently, a simple model-free controller called Time-Delay Estimation (TDE) was applied to CKCMs to solve the above issue. The TDE has been employed to control robot manipulators over the last decade because of its efficient computation capability [22; 23; 24]. It used time-delayed information to estimate unknown dynamics and disturbances in a sufficiently small time-delay. Lately, the TDE has been combined with Nonsingular Terminal Sliding Mode (NTSM) control [25] to provide highly robust and precise control schemes for robots with a fast convergence finite time. To our best knowledge, control schemes combining the TDE, the NFTSMC and synchronization have not been considered for controlling CKCMs.