## 4 Experimental Study  
This section presents an experimental study conducted to study the performance of the newly developed Syn-TDE-NFTSMC scheme in comparison with that of other existing control schemes when they are employed to control the motion of the manipulator. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Formula** \\ \hline \(\boldsymbol{\tau}(t)\in\mathds{R}^{2}\) & Joint force vector & \(\boldsymbol{\tau}(t)=\left(\tau_{1}\;\tau_{2}\right)^{T}\) \\ \(\boldsymbol{\tau}(t-L)\in\mathds{R}^{2}\) & Past joint force vector & \(\boldsymbol{\tau}(t-L)=\left(\tau(t-L)_{1}\;\tau(t-L)_{2}\right)^{T}\) \\ \(\mathbf{K}_{1},\;\mathbf{K}_{2},\mathbf{K},\;\text{and}\;\mathbf{K}_{sw}\) & \(2\times 2\) diagonal design matrices & \(\mathbf{K}_{1}=\begin{bmatrix}\mathbf{K}_{11}&0\\ 0&\mathbf{K}_{22}\end{bmatrix};\mathbf{K}_{2}=\begin{bmatrix}\mathbf{K}_{21}&0 \\ 0&\mathbf{K}_{22}\end{bmatrix}\) \\ \(p_{1},\;p_{2},\;q_{1},\;\text{and}\;q_{2}\) & Positive odd integers & \(1<p_{1}/q_{1}<2;1<p_{2}/q_{2}<2\) \\ \(\mathbf{u}\in\mathds{R}^{2}\) & Control law vector & \(\mathbf{u}=\left(\mathbf{u}_{1}\;\mathbf{u}_{2}\right)^{T}\) \\ \(\mathbf{s}\in\mathds{R}^{2}\) & Nonsingular terminal sliding surface vector & \(\mathbf{s}=\left(\mathbf{s}_{1},\;\mathbf{s}_{2}\right)^{T}\) \\ \(\text{sign}(\mathbf{s})\in\mathds{R}^{2}\) & Sign function of sliding surface vector & \(\text{sign}(\mathbf{s})=\left(\text{sign}(\mathbf{s}_{1}),\ldots,\text{sign}( \mathbf{s}_{\eta})\right)^{T}\) \\ \(\ddot{\mathbf{q}}_{-L}\in\mathds{R}^{2}\) & Past acceleration vector & \(\ddot{\mathbf{q}}_{(t-L)}=\left(\ddot{\mathbf{q}}_{(t-L)1};\ddot{\mathbf{q}}_{(t -L)2}\right)^{T}\) \\ \(\ddot{\mathbf{q}}_{d}\in\mathds{R}^{2}\) & Desired acceleration joint vector & \(\ddot{\mathbf{q}}_{d}=\left(\ddot{\mathbf{q}}_{d1},\ddot{\mathbf{q}}_{d2}\right) ^{T}\) \\ \(p_{1},\;p_{2},\;q_{1},\;\text{and}\;q_{2}\) & Positive odd integers & \(1<p_{1}/q_{1}<2;1<p_{2}/q_{2}<2\) \\ \(\boldsymbol{\alpha}\in\mathds{R}^{2\times 2}\) & Diagonal positive definite matrix & \(\boldsymbol{\alpha}=\begin{bmatrix}\boldsymbol{\alpha}_{2}&0\\ 0&\boldsymbol{\alpha}_{2}\end{bmatrix}\) \\ \(\mathbf{I}\in\mathds{R}^{2\times 2}\) & Identity matrix & \(\mathbf{I}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\) \\ \(\mathbf{D}\in\mathds{R}^{2\times 2}\) & Synchronization transformation matrix & \(\mathbf{D}=\begin{bmatrix}1&-1\\ -1&1\end{bmatrix}\) \\ \(\mathbf{e}\in\mathds{R}^{2}\) & Tracking error vector & \(\mathbf{e}=\left(\mathbf{e}_{1},\;\mathbf{e}_{2}\right)^{T}\) \\ \(\mathbf{e}_{s}\in\mathds{R}^{2}\) & Synchronization error vector & \(\mathbf{e}_{s}=\left(\mathbf{e}_{s1},\;\mathbf{e}_{s2}\right)^{T}\) \\ \(\mathbf{e}_{c}\in\mathds{R}^{2}\) & Cross-coupling error vector & \(\mathbf{e}_{c}=\left(\mathbf{e}_{c1},\;\mathbf{e}_{c2}\right)^{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The symbols, specifications, and parameters. In our experimental study, the newly developed Syn-TDE-NFTSMC scheme derived in (14), along with other existing control schemes described in Appendix A, were used to control the manipulator to track a circular trajectory over a duration of 45 s, specified by \(x_{des}(t)\) and \(y_{des}(t)\) as follows. \[\left\{ \begin{aligned} x_{des}(t)&=0.3683+0.0383\cos(2 \pi t/45)\\ y_{des}(t)&=-0.9968+0.0381\sin(2\pi t/45)\end{aligned} \right. \tag{20}\]  
In order to use the optimized control parameters obtained from our simulation study for our experimental study, we intentionally selected the above circular motion, which was the same motion we used in our simulation study. In addition, we wanted to compare the results of the experimental study with those of the simulation study for the same motion. In [15], computer simulation was performed to evaluate the performance of the newly developed Syn-TDE-NFTSMC scheme and several existing control schemes including LINEAR, TDE-based LINEAR, TDE-based LINEAR with synchronization errors, and TDE-based SMC with synchronization errors while tracking the same motion.