## 6 Conclusions  
This paper proposes an improved Sparrow Search Algorithm (ISSA) for optimizing the PID parameters of a bridge crane control system. Firstly, the traditional Sparrow Search Algorithm is enhanced by introducing Tent Map Chaos during the initialization phase of the algorithm iteration to enhance the quality of initial solutions. Additionally, the algorithm integrates the concepts of the sine-cosine algorithm and introduces a nonlinear learning factor to adjust the position updating mechanism of discoverers, thereby enhancing both local exploitation and global exploration capabilities. The Levy flight strategy is introduced to improve the follower's position updating method, enhancing its ability to escape local optima. Finally, to address overshoot in the trolley position of the crane, an overshoot penalty term is introduced into the fitness function. Comparative analysis with existing PSO, SA, WOA, and SSA algorithms demonstrates that the ISSA algorithm exhibits more precise search capability while meeting the convergence requirements.  
The application of the proposed algorithm to the control system revealed experimental results indicating that the overshoot of all five algorithms was less than 5%, underscoring the effectiveness of the improved fitness function. Furthermore, in comparison with other algorithms, the ISSA algorithm demonstrated superior performance in both trolley position control and effective suppression of payload swing angle. When subjected to varying working conditions and external disturbances, the ISSA algorithm exhibited the fastest response time and shortest steady-state time. These findings validate the robustness and high performance of the ISSA algorithm. Consequently, the PID controller optimized using the ISSA algorithm can be effectively employed for trolley position control and payload swing angle suppression in bridge crane systems, thereby satisfying the control requirements of such systems.  
Subsequent research will combine other control methods of the proposed ISSA, such as fuzzy control and neural networks, and apply them to more complex systems, such as crane double-pendulum systems, power generation, and so on.  
Conceptualization, Y.Z. and L.L.; methodology, Y.Z. and L.L.; software, Y.Z., L.L. and J.L.; validation, Y.Z., L.L., J.L., J.C., C.K. and D.H.; formal analysis, Y.Z. and L.L.; investigation, Y.Z. and L.L.; resources, Y.Z. and L.L.; data curation, Y.Z., L.L., J.L., J.C., C.K. and D.H.; writing--original draft, Y.Z. and L.L.; writing--review and editing, Y.Z., L.L., J.L. and C.K.; visualization, Y.Z. and L.L.; supervision, Y.Z. and L.L.; project administration, Y.Z. and L.L. All authors have read and agreed to the published version of the manuscript.  
Figure 24: Payload swing angle suppression effects of the five algorithms after input command variations.  
**Funding:** This research was funded by the University-Industry Cooperation Project \"Research and Application of Intelligent Traveling Technology for Steel Logistics Based on Industrial Internet\", grant number 2022H6005; Natural Science Foundation of Fujian Provincial Science and Technology Department, grant number 2022J01952; Research Start-up Projects, grant number GY-Z12079; Fujian University Industry-University Cooperation Science and Technology Programme, grant number 2022N5020.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
**Conflicts of Interest:** The authors declare no conflicts of interest.