## 6 Computer Simulation Study
### Simulation Results  
The control schemes listed in Table 2 are used in the computer simulation to control the end-effector of the manipulator to track a circle specified by \(x_{des}(t)\) and \(y_{des}(t)\) as follows. \[ \begin{cases}x_{des}(t)=0.3683+0.05\cos(\pi t+\pi/2)\\ y_{des}(t)=-0.4183-0.05\sin(\pi t/10+\pi/2)\end{cases}  \tag{52}\]  
The results obtained from the simulation are presented in Figures 5-9 and Tables 3-5. Figure 5 shows the planar motions of the manipulator end-effector when controlled by the above control schemes while Figure 6 presents the time trajectories of the tracking errors \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Robot Parameters** & **Description** & **Value** & **Unit** \\ \hline \(m\) & Link’s total mass & 4.91 & kg \\ \(m_{1}\) & Link’s moving part mass & 0.59 & kg \\ \(d\) & Ground’s horizontal distance & 0.74 & m \\ \(l_{s}\) & Link’s fixed length & 0.26 & m \\ \(\mathrm{F}_{V_{1}}\) & Viscous friction coefficient of the 1 st link & 5 & N-m/s/rad \\ \(\mathrm{F}_{V_{2}}\) & Viscous friction coefficient of the 2 nd link & 5 & N-m/s/rad \\ \(\mathrm{F}_{\mathrm{C}_{1}}\) & Coulomb friction coefficient of the 1 st link & 5 & N-m \\ \(\mathrm{F}_{\mathrm{C}_{2}}\) & Coulomb friction coefficient of the 2 nd link & 5 & N-m \\ \(\mathrm{g}\) & Gravitational acceleration constant & 9.81 & m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The robot parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Control Scheme** & **Control Parameters** \\ \hline LINEAR & \(\overline{\mathbf{M}}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{D}\)\(=\) diag(200, 200), \(\mathbf{K}_{P}\)\(=\) diag(7.704 \(\times 10^{3},7.704\times 10^{3}\)) \\ TDE-LINEAR & \(L=9.999\times 10^{-4}\)s, \(\overline{\mathbf{M}}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{D}\)\(=\) diag(200, 200), \(\mathbf{K}_{P}\)\(=\) diag(7.704 \(\times 10^{3},7.704\times 10^{3}\)) \\ Syn-TDE-LINEAR & \(L=9.999\times 10^{-4}\)s, \(\overline{\mathbf{M}}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{D}\)\(=\) diag(200, 200), \(\mathbf{K}_{P}\)\(=\) diag(7.704 \(\times 10^{3},7.704\times 10^{3}\)), \\ Syn-TDE-SMC & \(L=9.999\times 10^{-4}\)s, \(\overline{\mathbf{M}}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{1}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}\)\(=\) diag(10, 10), \(\mathbf{K}_{sw}\)\(=\) diag(5, 5), \\ & \(\boldsymbol{\alpha}\)\(=\) diag(0.5, 0.5) \\ Syn-TDE-NFTSMC & \(L=9.999\times 10^{-4}\)s, \(p_{1}=19\), \(p_{2}=11\), \(q_{1}=17\), \(q_{2}=9\), \(\overline{\mathbf{M}}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{1}\)\(=\) diag(25, 25), \\ & \(\mathbf{K}_{2}\)\(=\) diag(5, 5), \(\mathbf{K}\)\(=\) diag(0.1, 0.1), \(\mathbf{K}_{sw}\)\(=\) diag(15, 15), \(\boldsymbol{\alpha}\)\(=\) diag(0.5, 0.5) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the control schemes. \(\mathbf{e}(t)\) of the control schemes. Table 3 tabulates the absolute average tracking errors (AATE) of the control schemes computed by MATLAB. Figure 5: Trajectory tracking in X-Y plane (circular motion). Figure 6: Time trajectories of tracking errors of the control schemes, of joint 1 (**a**) and joint 2 (**b**). Figure 7: Time trajectories of synchronization errors and cross coupling errors of the control schemes, of joint 1 (**a**) and joint 2 (**b**). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tracking Errors** & **Syn-TDE-LINEAR** & **Syn-TDE-SMC** & **Syn-TDE-NFTSMC** \\ \hline \(e_{s_{1}}\) (mm) & 0.0206 & \(3.82\times 10^{-3}\) & \(2.16\times 10^{-3}\) \\ \(e_{s_{2}}\) (mm) & 0.0206 & \(3.82\times 10^{-3}\) & \(2.16\times 10^{-3}\) \\ \(e_{c_{1}}\) (mm) & 0.028 & \(1.03\times 10^{-2}\) & \(9.07\times 10^{-3}\) \\ \(e_{c_{2}}\) (mm) & 0.016 & \(5.33\times 10^{-3}\) & \(4.99\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The absolute average synchronization errors (AASE) and cross-coupling errors (AACE) of the control schemes, computed by MATLAB. Figure 8: Time trajectories of estimation errors of Syn-TDE-NFTSMC of joint 1 (**a**) and joint 2 (**b**).