## 5 Simulations
### Simulation Results and Discussion: Exploring Findings and Analysis  
The tracking accuracy is assessed using the root-mean-square (RMS) algorithm, and the resulting RMS errors are provided in Table 4. Each joint's RMS error is denoted as \(e_{1}\) (Joint 1), \(e_{2}\) (Joint 2), and \(e_{3}\) (Joint 3). Figure 3 demonstrates the performance of an RBFNN in approximating the dynamics of the robot and handling unknown uncertain terms. It is evident that by combining the benefits of RBFNN and PPCA, the system can quickly obtain accurate approximations for the entire robot dynamics and unknown uncertain terms. In Figure 4, the robot's end-effector trajectory is manipulated to closely track the desired path provided in Table 2, aiming for the highest possible accuracy. The model-based methods, including SMC, TSMC, and FTSMC, exhibit matching with the desired trajectory, showcasing their effectiveness. Additionally, the proposed model-free method also demonstrates close adherence to the desired trajectory. In contrast, the trajectory produced by the PID controller exhibits a noticeable deviation from the desired path, indicating a relatively lower level of accuracy. To thoroughly analyze and evaluate the effectiveness of each method, we rely on the simulation results presented in Figures 5-8 and Table 4. These results allow for a precise and specific examination of the performance of each method, providing valuable insights into their respective strengths and limitations. Upon closer examination of the smaller figures within Figures 5-7, which depict a zoomed-in view of the phase from 0 to 1 second, it becomes evident that only the developed method exhibits convergence within the prescribed performance. This indicates that the developed method achieves fixed-time convergence, whereas the other methods do not demonstrate the same level of convergence. \begin{table}
\begin{tabular}{c c c c c} \hline
**PID (\(\epsilon_{1,2,3}\))** & **SMC (\(\epsilon_{1,2,3}\))** & **TSMC (\(\epsilon_{1,2,3}\))** & **FTSMC (\(\epsilon_{1,2,3}\))** & **Developed Algorithm (\(\epsilon_{1,2,3}\))** \\ \hline \(1.4\times 10^{-3}\) & \(1.890\times 10^{-4}\) & \(2.475\times 10^{-5}\) & \(2.383\times 10^{-5}\) & \(7.2\times 10^{-7}\) \\ \hline \(3.45\times 10^{-2}\) & \(3.376\times 10^{-4}\) & \(4.672\times 10^{-5}\) & \(3.941\times 10^{-5}\) & \(1.539\times 10^{-5}\) \\ \hline \(7.4\times 10^{-3}\) & \(3.502\times 10^{-4}\) & \(5.264\times 10^{-5}\) & \(5.030\times 10^{-5}\) & \(1.64\times 10^{-6}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of the control performance from five controllers for trajectory tracking. Furthermore, when analyzing the smaller figures within Figures 5-7, which depict a zoomed-in view of the phase from the 2 nd to 20 th second, it becomes apparent that the PID controller fails to deliver satisfactory tracking performance during normal operations. This can be attributed to its inability to handle uncertainties and disturbances, which are  
Figure 4: Analyzing the effects of different controllers on the end-effector trajectory of the robot. Figure 3: Performance of an RBFNN in approximating robot dynamics and unknown uncertain terms. commonly present in the system. The PID controller exhibits large tracking errors in the presence of uncertainties and disturbances. On the other hand, the SMC, TSMC, and FTSMC controllers demonstrate improved tracking performances with steady-state errors within predefined bounds. Although the SMC controller handles uncertainties and disturbances better than the PID controller, its performance still falls short of the desired level when compared to the TSMC and FTSMC controllers. In contrast, despite the influence of unknown robot dynamics and uncertain terms, the proposed controller, designed appropriately, yields the smallest steady-state errors among all the methods considered. Figure 5: Analyzing the temporal variation of the tracking error in the first joint of a robot manipulator.