## 4 Simulation
### Two-DOF Robotic Arm Simulation
Through multiple iterations and by simulating different controller parameter settings, the corresponding integral of time-weighted absolute error (ITAE) values is calculated. The parameter combination that minimizes the ITAE is selected to achieve the optimization of the overall system. From Figures 7a and 8a, it is observed that the first arm reaches a steady state in 0.224 s while tracking the desired trajectory, with an overshoot of \(3.1004\times 10^{-1}\) rad. The steady state error is \(9.86\times 10^{-5}\) rad for the first 5 s, and \(9.68\times 10^{-5}\) rad for the 5 to 10 s interval. From Figures 7b and 8b, it is noted that the second arm reaches a steady state in 0.279 s while tracking the desired trajectory, with an overshoot of \(2.0075\times 10^{-1}\) rad. The steady state error is \(1.075\times 10^{-5}\) rad for the first 5 s, and \(1.076\times 10^{-5}\) rad for the 5 to 10 s interval. Figure 4: The simulation of the trajectory tracking without the ISSA. (**a**) The simulation results of the first link. (**b**) The simulation results of the second link. Figure 5: The simulation of the trajectory tracking error without the ISSA. (**a**) The simulation results of the first link. (**b**) The simulation results of the second link. To verify its robustness, we conducted additional comparative simulations using the PD control algorithm alongside the fixed-time control (FTC) algorithm, with the latter optimized by the ISSA (improved sparrow search algorithm). By comparing the tracking error trajectories of both control methods across two joints, we obtained a more comprehensive understanding of the control performance. The PD control is given as follows:  
\[\tau_{i}=-K_{P}(q_{i}-q_{id})-K_{D}\dot{q}_{i},i=1,2 \tag{59}\]  
where \(K_{P}\) and \(K_{D}\) represent the proportional gain and differential gain, respectively. The control gains are set to \(K_{P}=43\) and \(K_{D}=11.8\). Figure 9 presents the corresponding results of the simulations. Figure 6: Control torques: (**a**) the simulation results of the first link; (**b**) the simulation results of the second link. Figure 7: The simulation of trajectory tracking with the ISSA. (**a**) The simulation results of the first link. (**b**) The simulation results of the second link. By comparing Figures 9a and 10a, it can be observed that at \(3\) s, the trajectory errors of the first link of the robotic arm are \(-2.59\times 10^{-3}\)rad and \(-9.87\times 10^{-5}\)rad, respectively, while at \(8\) s, the errors are \(-9.26\times 10^{-4}\)rad and \(-9.68\times 10^{-5}\)rad. Similarly, from the comparison between Figures 9b and 10b, it can be observed that at \(3\) s, the trajectory errors of the second link of the robotic arm are \(-2.83\times 10^{-4}\)rad and \(-1.08\times 10^{-5}\)rad, respectively, while at \(8\) s the errors are \(-1.03\times 10^{-4}\)rad and \(-1.08\times 10^{-5}\)rad. It is evident that the steady-state error of the robotic arm under the fixed-time control combined with ISSA is smaller than that under PD control. To further demonstrate the robustness of the FTC with ISSA, we introduced external disturbances to the two-DOF robotic arm and compared its performance with that of the PD control. This comparison highlights the superior robustness of the FTC-ISSA method under external interference conditions, as reflected in the error convergence and tracking accuracy metrics. To simulate potential external disturbances that the system might encounter in real-world operations, a random external disturbance force \(F_{d}\) was applied during the \(3\) to \(4\) s interval to evaluate the robustness and stability of the control system. The disturbance force acts on both joints of the robotic arm, with each component affecting one of the two degrees  
Figure 8: The simulation of trajectory tracking errors with the ISSA. (**a**) The simulation results of the first link. (**b**) The simulation results of the second link.