## 5 Simulations
### Testing System Configuration: A Detailed Examination of the Setup
In the TSMC approach [46], a sliding function is constructed as follows:  
\[S=e_{1}+\mathcal{C}^{-1}\text{Sig}^{\frac{p}{4}}(e_{2})\, \tag{48}\]  
where \(\mathcal{C}=\text{diag}(c_{i})\) with \(c_{i}>0\) for \(i=1,\ldots,n\), and \(p\) and \(q\) (where \(p>q\)) are positive odd numbers. The control torque obtained from the output of the TSMC is expressed as follows:  
\[\tau=\tilde{M}(\alpha_{1})\bigg{(}\dot{\alpha}_{2d}+\mathcal{C}\frac{q}{p}e_ {2}^{2-\frac{p}{4}}+\Lambda\text{sign}(S)+\Delta S\bigg{)}+\tilde{C}(\alpha_ {1},\alpha_{2})\alpha_{2}+\tilde{G}(\alpha_{1}) \tag{49}\]  
where \(\Delta=\text{diag}(\delta_{i})\) with \(\delta_{i}>0\) for \(i=1,\ldots,n\), and \(|\mathcal{F}|\leq\Lambda\). \begin{table}
\begin{tabular}{c c} \hline
**Terms** & **Described Function** \\ \hline \multirow{4}{*}{Desired Trajectory (m)} & \(\text{X}_{d}=0.85-0.01t\) \\  & \(\text{Y}_{d}=0.2+0.2\sin(\frac{t}{2})\) \\  & \(Z_{d}=0.7+0.2\cos(\frac{t}{2})\) \\ \hline \multirow{4}{*}{Friction Forces and Disturbances \(\mathcal{F}_{r}(\alpha_{2})+d(t)\) (N\(\cdot\)m)} & \(\mathcal{F}_{r3}(\alpha_{2})+d_{3}(t)=0.1\text{sign}(\alpha_{23})+2\alpha_{23}+ 6\sin(t)\) \\  & \(\mathcal{F}_{r}(\alpha_{2})+d_{3}(t)=0.1\text{sign}(\alpha_{23})+2\alpha_{23}+ 6\sin(t)\) \\ \hline \end{tabular}
\end{table} 
Table 2: Investigating the effectiveness of a proposed control system by tracking the desired trajectory for a robot manipulator with assumed uncertain terms in the simulation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Technique** & **Notation** & **Value** & & & & & & & & \\ \hline PPC & \(\ell_{0}\), \(\ell_{1}\), \(\ell_{\infty}\), \(\phi\) & \([\ \ 0.34\) & \(0.15\) & \(0.3\ \ ]\), \(0.06\), \(0.001\), \(6\) & & & & & & \\ \hline \multirow{4}{*}{RBFNN} & \(\mathcal{K}_{N}\), \(\varphi_{3}\), \(\varphi_{4}\), \(\mathcal{W}_{0}\) & \multicolumn{6}{c}{\(\frac{1}{1000^{\prime}}\), \(0.001\), \(0.001\), \(\text{zeros}(9,3)\)} \\  & \(\sigma\), \(l\), \(\mathcal{D}\) & \(0.01\times\left[\begin{array}{ccccccccc}-2&-1.5&-1&-0.5&0&0.5&1&1.5&2\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ -2&-1.5&-1&-0.5&0&0.5&1&1.5&2\\ \hline \multirow{2}{*}{Developed Algorithm} & \(\kappa_{1}\), \(\kappa_{2}\), \(\xi_{1}\), \(\xi_{2}\) & \(0.1\), \(0.1\), \(1.4\), \(0.8\) & & & & & & \\  & \(\psi_{1}\), \(\psi_{2}\), \(\alpha_{1}\), \(\omega_{2}\), \(\gamma^{*}\) & \(10\), \(500\), \(1.2\), \(0.8\), \(0.00001\) &In the FTSMC approach [47], a sliding function is constructed as follows:  
\[S=e_{2}+C\text{Sig}^{\xi}(e_{1})+\omega\text{Sig}^{\beta}(e_{1}) \tag{50}\]  
where \(\mathcal{C}=\text{diag}(c_{i})\) with \(c_{i}>0\), \(\omega=\text{diag}(\omega_{i})\) with \(\omega_{i}>0\), \(\xi\) and \(\beta\) are parameter vectors with elements \(\zeta_{i}>1\) and \(0<\beta_{i}<1\), respectively, for \(i=1,\ldots,n\). The control torque obtained from the output of the FTSMC is expressed as follows:  
\[\tau=\bar{M}(\alpha_{1})\big{(}\dot{\alpha}_{2d}+\mathcal{Z}+\Lambda\text{ sign}(S)+\Delta S\big{)}+\bar{C}(\alpha_{1},\alpha_{2})\alpha_{2}+\bar{G}( \alpha_{1}) \tag{51}\]  
where \(\Delta=\text{diag}(\delta_{i})\) with \(\delta_{i}>0\), \(\mathcal{Z}=[z_{1},\ \ldots,\ z_{n}]\) with \(z_{i}=\zeta_{i}c_{i}|e_{i}|^{\zeta_{i}-1}\ell_{i}+\beta_{i}\omega_{i}|e_{i}|^{ \bar{\delta}_{i}-1}e_{i}\), for \(i=1,\ldots,n\), and \(|\mathcal{F}|\leq\Lambda\). Assuming we do not have knowledge of all the robot dynamics, including \(\bar{M}(\alpha_{1})\), \(\bar{C}(\alpha_{1},\alpha_{2})\), \(\bar{G}(\alpha_{1})\), and the uncertain term \(\mathcal{F}\), we have two options for controlling the robot: the PID controller or the developed algorithm. The control torque obtained from the output of the PID [5] is expressed as follows:  
\[\tau=-K_{P}e_{1}-K_{I}\int e_{1}-K_{D}e_{2} \tag{52}\]  
where the control gains \(K_{P},K_{I}\), and \(K_{D}\) are positive.