## 1 Introduction  
Closed-Kinematic Chain Manipulators (CKCMs) have drawn significant attention in the robotics community due to their ability to achieve precise Cartesian motion through coordinated active joint movements. Compared to Open-Kinematic Chain Manipulators(OKCM), CKCMs offer improved positioning accuracy and payload handling capabilities [1; 2; 3; 4]. However, CKCMs also present challenges, including limited workspace, absence of closed-form solutions for forward kinematics, and synchronization issues among joints and end-effectors. Since synchronization errors can degrade tracking accuracy and even damage the mechanical structure, especially under high accelerations or abrupt payload changes, various synchronization-based control schemes have been developed [5; 6; 7; 8; 9]. To implement effective control schemes in a real robot in real time, the control laws must have simple structures and should not rely on knowledge of the manipulator's dynamics. As a result, synchronized adaptive and PD control schemes have been explored in [10; 11; 12] for this purpose. Additionally, a traditional sliding mode control (SMC) scheme developed by authors in [13; 14] was employed to control the motion of CKCMs. SMC is advantageous due to its robustness against uncertainties and disturbances in system parameter variations, making it suitable for CKCMs [13; 15; 16; 17]. However, the SMC approach applied to manipulators can only guarantee asymptotic stability, requiring finite time to converge to an equilibrium. To achieve finite-time convergence, the terminal SMC (TSMC) scheme was developed in [18; 19; 20]. This approach includes a driving component that forces the system's trajectory towards a stable sliding surface, which is designed to enforce the system's desired error dynamics. However, TSMC has two main limitations: a slower convergence to equilibrium as compared to the traditional SMC when the system state is far from the equilibrium, and an issue in dealing with singularity [21; 22]. To address these, an enhanced version called Nonsingular Fast Terminal Sliding Mode Control (NFTSMC) was introduced in [23; 24], incorporating synchronization to control a parallel robot manipulator. This improved control scheme overcomes singularity and achieves fast convergence [25]. Despite its advantages, the gain selection in NFTSMC still depends on conservative estimates of the manipulator's dynamic model, resulting in calculational challenges due to the highly complicated dynamic model of the manipulator. To overcome this issue, a model-free controller utilizing Time Delay Estimation (TDE) was developed. Over the past decade, TDE-based controllers have been widely used for robot manipulators due to their computation efficiency [26; 27; 28]. By using time-delayed information, TDE effectively estimates unknown dynamics and disturbances within a sufficiently small time delay, where the uncertainty can be considered as a continuous or piecewise continuous function. Thus, the approximation uncertainty is inherently satisfied [24]. In addition, TDE has been combined with Nonsingular Terminal Sliding Mode (NTSM) control [29] to deliver highly robust and precise control with fast, finite-time convergence. A recent study [15] introduced Syn-TDE-NFTSMC for CKCMs, a TDE-based NFTSMC scheme with synchronization error handling. A simulation study indicated that this control scheme outperformed existing control schemes, including LINEAR, TDE-based LINEAR (TDE-LINEAR), TDE-based LINEAR with synchronization errors (Syn-TDE-LINEAR), and TDE-based SMC with synchronization errors (Syn-TDE-SMC), in controlling the CKCM motion. For instance, a comparison of the tracking errors showed that Syn-TDE-NFTSMC achieved a better performance, with the error of \(4.07\times 10^{-3}\) mm for Joint 1 being notably smaller than those of the other control schemes--specifically, errors of 0.32 mm, 0.0197 mm, 0.0197 mm, and \(7.17\times 10^{-3}\) mm were observed for LINEAR, TDE-LINEAR, Syn-TDE-LINEAR, and Syn-TDE-SMC, respectively.