## 3 The Control Scheme  
The Syn-TDE-NFTSMC scheme developed in [15] is now applied to control the motion of the manipulator described above in Section 2. As shown in Figure 4, the control system developed for the manipulator mainly consists of three subsystems: the Synchronization Subsystem, the NFTSMC Subsystem, and the TDE Subsystem. The operation of the control system can be briefly explained as follows. As the input of the control scheme, the Cartesian vector \(\mathbf{x}_{d}\in\mathbb{R}^{2}\) representing the desired manipulator's Cartesian configuration (position and orientation) is either specified by the  
Figure 4: The Syn-TDE-NFTSMC scheme used to control the 2-DOF CKCM. user or generated by a trajectory planner. The vector \(\mathbf{x}_{d}\) is then converted into its corresponding joint vector \(\mathbf{q}_{d}\in\mathbb{R}^{2}\) through the Inverse Kinematic Transformation specified by Equations (1) and (2). The actual joint variables \(q_{1}\) and \(q_{2}\) acquired by the LVDTs constitute the actual joint vector \(\mathbf{q}\in\mathbb{R}^{2}\). Then, both the desired joint vector \(\mathbf{q}_{d}\) and the actual joint vector \(\mathbf{q}\) are sent to the Synchronization Subsystem. This subsystem calculates the position error for each active joint \(\mathbf{e}_{i}=\mathbf{q}_{d_{i}}-\mathbf{q}_{i}\) for Joint 1 and Joint 2, the synchronization error \(\mathbf{e}_{S}\), and the cross-coupling error \(\mathbf{e}_{c}\) between the two joints. The cross-coupling error \(\mathbf{e}_{c}\in\mathbb{R}^{2}\) is then processed by NFTSMC, which defines a sliding surface to regulate error behavior. Next, the control law \(\mathbf{u}\in\mathbb{R}^{2}\) is formulated based on this sliding surface, ensuring that the system trajectory converges towards a stable hyperplane (sliding surface). This control law then serves as a key component of the input \(\boldsymbol{\tau}\) for the actuators of the manipulator to ensure asymptotic convergence of both position and synchronization errors in minimal time. Additionally, the TDE Subsystem utilizes past control inputs \(\boldsymbol{\tau}(t-L)\in\mathbb{R}^{2}\) and the acceleration data \(\widetilde{\mathbf{q}}_{t-L}\in\mathbb{R}^{2}\) with \(\widetilde{\mathbf{M}}(\mathbf{q})\in\mathbb{R}^{2\times 2}\), a constant, selected diagonal matrix of the estimated time delay \(L\) to estimate all nonlinear terms' system dynamics \(\widetilde{\mathbf{H}}\in\mathbb{R}^{2}\), which include the inertia uncertainty, Coriolis/centripetal vector, gravitational vector, friction vector, disturbances, and disturbance torques \(\boldsymbol{\tau}_{d}\in\mathbb{R}^{2}\). These estimates are essential for generating the control input applied to the manipulator, ensuring robust and precise control performance under varying dynamic conditions. We proceed to implement the Syn-TDE-NFTSMC scheme developed in [15] to control the motion of the manipulator. First the dynamics of the manipulator specified by (5) can be simplified as follows:  
\[\widetilde{\mathbf{M}}\widetilde{\mathbf{q}}+\mathbf{H}(\mathbf{q}\dot{ \mathbf{q}},\widetilde{\mathbf{q}})= \boldsymbol{\tau} \tag{12}\]  
where \(\mathbf{H}(\mathbf{q}\dot{\mathbf{q}},\ddot{\mathbf{q}})\), including the manipulator dynamics with disturbance torques, can be denoted as follows:  
\[\mathbf{H}(\mathbf{q}\dot{\mathbf{q}},\ddot{\mathbf{q}})=[\mathbf{M}(\mathbf{ q})-\widetilde{\mathbf{M}}]\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{ \mathbf{q}})\dot{\mathbf{q}}+\mathbf{G}(\mathbf{q})+\mathbf{F}(\mathbf{q}, \dot{\mathbf{q}})+\boldsymbol{\tau}_{d} \tag{13}\]  
Based on Figure 4 and according to [15], by using the TDE method, the nonlinear manipulator dynamic term \(\mathbf{H}(\mathbf{q}\dot{\mathbf{q}},\ddot{\mathbf{q}})\) can be approximated by the estimated term \(\widetilde{\mathbf{H}}(\mathbf{q}\dot{\mathbf{q}},\ddot{\mathbf{q}})\) within a minimum time \(L\), and a control law \(\mathbf{u}\) can be added to the input \(\boldsymbol{\tau}\) to improve tracking accuracy and enhance robustness.