## 6 Computer Simulation Study
### Simulation Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**AAEE** & **Syn-TDE-NFTSMC** \\ \hline \(e_{est_{1}}\) (Nm) & 0.0177 \\ \(e_{est_{2}}\) (Nm) & 0.0172 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The absolute average estimation errors (AAEE) of the Syn-TDE-NFTSMC computed by MATLAB. Figure 9: The control input signals of the control schemes of joint 1 (**a**) and joint 2 (**b**). Figure 7 presents the time trajectories of the synchronization errors (\(\mathbf{e}_{s}(t)\)) and cross-coupling (\(\mathbf{e}_{c}(t)\)) of the control schemes while Figure 8 presents the estimation errors (\(\mathbf{e}_{est}(t)\)) of the Syn-TDE-NFTSMC. Table 4 tabulates the absolute average synchronization errors (AASE) and cross-coupling errors (AACE) of the control schemes while Table 5 tabulates the absolute average estimation errors (AAEE) of the Syn-TDE-NFTSMC, computed by MATLAB. The control inputs of both joints show no chattering, as shown in Figure 9a,b. From Figures 5-7, it is seen that Syn-TDE-LINEAR provided better performance and faster error convergence than both LINEAR and TDE-LINEAR. It is meaningful that the involvement of TDE and the synchronization errors improved the performance of the control schemes. Syn-TDE-SMC showed a better tracking path than the Syn-TDE-LINEAR. Finally, we see that our proposed control scheme, Syn-TDE-NFTSMC, was on track with the fastest desired trajectory with the slightest deviation (from the desired path in Figure 5) and had the fastest error convergence compared to other existing control schemes. From Table 3, based on the computed AATEs of the control scheme, it is clear that inclusion of TDE and the synchronization errors improved the performance of control schemes as for example the AATE of the TDE-LINEAR (0.0197 mm) is smaller than that of the LINEAR (0.32 mm). Other AATEs in the table validate the above observation. From the results presented in Table 3, we see that our proposed control scheme, namely Syn-TDE-NFTSMC, has the best tracking performance as compared to other existing control schemes due to its smallest AATEs for both joint variables. From Table 4, based on the computed AASEs and AACEs of the control scheme, it is clear that inclusion of Syn-TDE-NFTSMC improved the performance of control schemes as for example the AASEs of the Syn-TDE-NFTSMC (\(2.16\times 10^{-3}\) mm) is smaller than that of the Syn-TDE-SMC (\(3.82\times 10^{-3}\) mm). Other AASEs and AACEs in the table validate the above observation. From the results presented in Table 4, we see that our proposed control scheme has the best tracking performance as compared to other existing control schemes due to its smallest AASEs and AACEs for both joint variables. Figure 8a,b plotted the nonlinear term \(\mathbf{H}_{i}=\mathbf{H}(\mathbf{q}\mathbf{\dot{q}}\mathbf{\ddot{q}})\), the estimation term \(\mathbf{h}_{i}=\mathbf{\tau}_{t-L}-\mathbf{\ddot{M}}\mathbf{\ddot{q}}_{t-L}\), and the estimation error \(\mathbf{e}_{est_{i}}=\mathbf{H}_{i}-\mathbf{h}_{i}=\mathbf{H}(\mathbf{q} \mathbf{\dot{q}}\mathbf{\ddot{q}})-\left(\mathbf{\tau}_{t-L}-\mathbf{\ddot{M }}\mathbf{\ddot{q}}_{t-L}\right)\), respectively, of the \(i\)th active joint. It can be seen that the estimation error remains close to zero. Furthermore, from the results presented in Table 5, we see that our proposed control scheme, namely Syn-TDE-NFTSMC, has the AAEEs close to zero as 0.0177 Nm and 0.0172 Nm for both joint variables. This implies that the TDE cancels the uncertainty, and the chattering phenomenon is reduced while maintaining the tracking accuracy. Therefore, it can be concluded that the proposed control scheme shows high-accuracy tracking performance with the model-free control performance in comparison with the other control schemes.