## 4 Simulation Analyses
In Stage 1 (from 0 s to 15 s), three of the control systems give similar good path tracking performance. In Stage 2 (from time greater than 15 s) and in Stage 3 (from time greater than 25 s), it is clear that the classical SMC provides the poorest path tracking performance, where robot operation becomes unstable when a large disturbance or uncertainty is applied. From Table 2 and Figure 4, it is observed that NFTSMC provides less path tracking error and faster transient response than classical SMC. However, tracking performance is also diminished upon application of a large disturbance. It is noteworthy that the proposed sliding surface is designed based on the sliding function integral in Equation (8), and this integral portion has a significant role in providing fast transient response and robustness against uncertainty and disturbances. Therefore. the proposed control strategy gives the best path tracking performance and fastest transient response among the compared control strategies, due to the role of the proposed surfaces, an adaptive compensator, and a main contribution of the proposed controller. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Control Strategy** & **Control Parameters** & **Parameter Value** \\ \hline Classical SMC & \(\eta,\Phi_{2},\rho_{2}\) & 2, 9.9, 1 \\ \hline NFTSMC & \(d,h,\beta\) & \(5,3,2\) \\  & \(\phi_{3},\rho_{3},\
u\) & 9.9, 1, 0.1 \\ \hline \multirow{3}{*}{Proposed Control Strategy (ANNFFSMC)} & \(h_{1},h_{2}\) & \(diag(10,10,10),diag(6,6,6)\) \\  & \(\Gamma_{1},\Gamma_{2},\Gamma_{3}\) & \(diag(3,3,3),diag(3,3,3),diag(2,2,2)\) \\ \cline{1-1}  & \(\theta,\alpha\) & \(diag(0.4,0.4,0.4),diag(1.2,1.2,1.2)\) \\ \cline{1-1}  & \(\gamma,\rho_{1},\varpi,\varepsilon^{*}\) & \(0.5,0.1,0.01,0.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The control parameter selection for the varying control strategies. SMC = sliding mode controller; ANNFFSMC = adaptive neural non-singular fast-terminal sliding-mode control. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Error Control Strategy** & \(\mathbf{E}_{1}^{\text{alg}}\) & \(\mathbf{E}_{2}^{\text{alg}}\) & \(\mathbf{E}_{3}^{\text{alg}}\) \\ \hline SMC & 0.1943 & 0.8708 & 0.0060 \\ NFTSMC & 0.1542 & 0.1218 & 0.0038 \\ ANNFFSMC & 0.0031 & 0.0031 & 0.0029 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The averaged tracking errors under control input signals of the control strategy. Figure 3: Trajectory tracking positions: (**a**) at Joint \(1\), (**b**) at Joint \(2\), and (**c**) at Joint \(3\). The control input signals for all control types, including classical SMC, NFTSMC, and the suggested system are shown in Figure 5. In Figure 5a, it is clear that the NFTSMC offers a continuous control signal by using a boundary technique [35]. However, the weakness of this technique is that a choice must be made between chattering phenomenon removal and path tracking precision. Consequently, this technique decreases the robustness of the system while also increasing the tracking error. In Figure 5b, the SMC offers a discontinuous control signal with serious chattering behavior. On the contrary, the suggested system offers a continuous control signal for the robot manipulator without the loss of its effectiveness, as shown in Figure 5c. The adaptations of the estimated parameters are shown in Figure 6. These adaptive gains are estimated according to the variation of the influences of disturbances and uncertainties, and they will attain a constant value once the error variables converge to the sliding surface in a stable phase. From the simulation performance, we conclude that the proposed controller gives the best performance compared to a classical SMC and NFTSMC in terms of tracking precision, transient response, chattering deletion, and small steady state error. Figure 4: Trajectory tracking errors: (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3.