## 5 Conclusions  
This paper presents a novel approach to the use of the integrated fixed-time control (FTC) algorithm and the improved sparrow search algorithm (ISSA) to minimize the trajectory tracking errors of robotic arms. By integrating the improved ISSA and the FTC, we presented rapid convergence and precise trajectory tracking performance under stringent constraints. The application of barrier Lyapunov functions (BLFs) has proven effective in ensuring the robustness and stability of the control system against disturbances and model uncertainties. While higher DOF systems offer more comprehensive validation, the proposed control algorithms are scalable and adaptable, and we expect similar performances in 3-DOF and 6-DOF systems. Future work will focus on extending this approach to more complex robotic arms in order to further validate its robustness and real-world applicability. A computer simulation for the two-DOF robotic arm was performed and we confirmed that the proposed hybrid approach significantly reduced tracking errors and improved response speeds compared with the traditional FTC algorithm alone. According to the chapter on \"Simulation on ISSA Performance\", it is evident that ISSA exhibits excellent optimization capabilities and robustness for the system. From the section on \"Two-DOF Robotic Arm Simulation\", it is understood that integrating ISSA with fixed-time control yields relatively precise controller parameters, achieving the objectives of reducing errors and enhancing response speed. Compared with traditional PD control methods, its superior robustness has been verified. Our simulation results confirm that there is an enhanced control performance, showcasing significant reductions in tracking \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time (s)**} & **Error of First Link with** & **Error of Second Link with Fixed-Time Control (rad)** & **Error of First Link with Fixed-Time Control + ISSA (rad)** & **Error of Second Link with Fixed-Time Control + ISSA (rad)** \\ \hline
3 & \(3.59\times 10^{-3}\) rad & \(3.92\times 10^{-4}\) rad & \(9.87\times 10^{-5}\) rad & \(9.68\times 10^{-5}\) rad \\
7 & \(1.97\times 10^{-3}\) rad & \(2.20\times 10^{-4}\) rad & \(9.68\times 10^{-5}\) rad & \(1.08\times 10^{-5}\) rad \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error summary table. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Time (s)**} & **Error of First Link with Fixed-Time Control (rad)** & **Error of Second Link with Fixed-Time Control + ISSA (rad)** & **Error of Second Link with Fixed-Time Control + ISSA (rad)** \\ \hline From 0 & \multirow{2}{*}{\(1.07\times 10^{-2}\) rad} & \multirow{2}{*}{\(8.87\times 10^{-4}\) rad} & \multirow{2}{*}{\(1.004\times 10^{-2}\) rad} & \multirow{2}{*}{\(7.50\times 10^{-4}\) rad} \\ to 5 s & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Overshoot summary table. errors and faster system responses. This hybrid approach not only leverages the quick convergence properties of fixed-time control but also capitalizes on the global search capabilities of ISSA, finetuning the control actions in order to adapt them dynamically to changing conditions. This study contributes to the field by demonstrating a scalable and reliable control strategy that enhances the operational efficiency of robotic arms in complex environments, paving the way for broader applications in advanced robotic systems. Conceptualization, M.T.V. and H.-S.C.; methodology, R.Z. ; software, D.J. ; validation, M.T.V., R.Z., H.C., P.H.N.A. and H.-S.C.; formal analysis, R.Z. ; investigation, H.-S.C. and M.T.V. ; resources, D.J. and R.Z. ; data curation, H.C. and P.H.N.A. ; writing--original draft preparation, R.Z., H.-S.C. and M.T.V. ; writing--review and editing, R.Z. ; supervision, H.-S.C. and M.T.V. ; project administration, H.-S.C.; funding acquisition, H.-S.C. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.