## 4 Experimental Study
Based on the simulation results, it was concluded that the control schemes that incorporated TDE and synchronization errors showed a better performance than conventional control schemes. Of all of the control schemes, Syn-TDE-NFTSMC delivered the desired trajectory with the smallest deviation from the desired motion and the fastest error convergence. Thus, in this experimental study, we compared the performance of Syn-TDE-NFTSMC with two existing TDE-based control schemes with synchronization errors that were evaluated in the simulation, namely TDE-based LINEAR with synchronization errors (Syn-TDE-PID) and TDE-based SMC with synchronization errors (Syn-TDE-SMC), when tracking the same motion for a 2-DOF-CKCM. The optimal control parameters obtained in the computer simulation [15] for the best tracking performance served as the basis for selecting the actual control parameters of the three control schemes in our experimental study. In the experiments, since our control schemes are model-free, tuning the parameters was quite straightforward. The parameters of the control schemes were tuned as follows. Since \(\mathbf{\widetilde{M}}\) and \(\mathbf{\alpha}\) are diagonal matrices, these parameters could be tuned by increasing the diagonal elements from small positive values while checking the control performance by trial and error. Other parameters including \(p_{1}\), \(p_{2}\), \(q_{1}\), \(q_{2}\), \(\mathbf{K_{1}}\), \(\mathbf{K_{2}}\), and \(\mathbf{K}\) were selected as described in [32] and \(\mathbf{K_{sw}}\) was selected as described in [15]. After conducting numerous experiments with all three control schemes, the most optimal control parameters were finalized and are tabulated in Table 4.  
The results of the experimental study are shown in Figures 5-9 and Tables 5-7. These results were used for comparison with those of the computer simulation study conducted in [15]. In general, due to experimental conditions and real-world factors, such as sensor  
\begin{table}
\begin{tabular}{c c} \hline
**Control Schemes** & **Control Parameters** \\ \hline Syn-TDE-PID & \(L=9.999\times 10^{-4}\) s, \(\mathbf{\widetilde{M}}=\text{diag}(0.9,\,0.9),\mathbf{K}_{p}=\text{diag}(50,\,4 5),\mathbf{K}_{i}=\text{diag}(0.4,\,0.5),\mathbf{K}_{\text{D}}=\text{diag}(3,\, 2.5),\) \\  & \(\mathbf{\alpha}=\text{diag}(0.5,\,0.5)\) \\ Syn-TDE-SMC & \(L=9.999\times 10^{-4}\) s, \(\mathbf{\widetilde{M}}=\text{diag}(0.3,\,0.25),\) \\  & \(\mathbf{\alpha}=\text{diag}(0.5,\,0.5)\) \\ Syn-TDE-NFTSMC & \(L=9.999\times 10^{-4}\)s, \(p_{1}=19,\,p_{2}=11,\,q_{1}=17,\,q_{2}=9,\,\mathbf{\widetilde{M}}=\text{diag }(0.3,\,0.25),\) \\  & \(\mathbf{K_{1}}\)\(=\text{diag}(25,\,25),\) \(\mathbf{K_{2}}=\text{diag}(20,\,20),\)K\(=\text{diag}(9,\,9),\) \(\mathbf{K}_{sw}=\text{diag}(8,\,12),\)\(\mathbf{\alpha}=\text{diag}(0.5,\,0.5)\) \\ \hlinenoise, environmental disturbances, actuator backlash, and mechanical limitations, the experimental results showed slightly higher tracking errors as compared to the computer simulation results. The details are presented below. Figure 5: Trajectory tracking in the X-Y planes of the control schemes used in the experimental study (circular motion). Figure 6: The time trajectories of the tracking errors of the control schemes of Joint 1 (**a**) and Joint 2 (**b**) in the experimental study. Figure 8: The time trajectories of the estimation errors of the control schemes of Joint 1 (**a**,**c**) and Joint 2 (**b**,**d**) in the experimental study. Figure 7: The time trajectories of the synchronization errors and cross-coupling errors of the control schemes of Joint 1 (**a**) and Joint 2 (**b**) in the experimental study. Figures 5-9 indicate that incorporating TDE and synchronization errors greatly enhanced the performance of the control schemes. Our developed control scheme, Syn-TDE-NFTSMC, closely followed the desired trajectory (as seen in Figure 5) with minimal deviation from the desired path and demonstrated rapid error convergence.