# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 6 Conclusions  
From the perspective of the control effect, the simulation results and relevant data presented above demonstrate that the CGSA algorithm is effective for the automatic U-turn control of ships in restricted waters. The introduction of nonlinear feedback does not compromise the controller's performance. The figures show that, under favorable port conditions, a ship can complete the automatic U-turn operation in a restricted water area within 10 min, with the duration of large rudder angles controlled to approximately 1-3 min. After incorporating the nonlinear feedback link into the original control system, both the time spent at large rudder angles and the range of the U-turn are reduced to some extent, and the average rudder angle of the ship is also decreased throughout the turning process. Therefore, the theoretical proof and simulation experiments conducted above confirm that the introduction of a nonlinear feedback link in the control system can achieve the same control effect with less energy input, ensure the smooth output of various variables, reduce the wear of the steering gear, extend its service life, and effectively realize energy savings during navigation. The reduction in energy consumption not only lowers operational costs but also aligns with the initiatives to decarbonize the shipping industry; it holds significant engineering implications for the motion control of large vessels that are challenging to maneuver and respond slowly. Our findings directly contribute to the goals of green shipbuilding and operational efficiency. Based on the current achievements, future research will integrate advanced intelligent control algorithms to further optimize the study of ship motion control in port areas; furthermore, we will improve the current model, taking into account the nonlinear and inertial characteristics of ships under harsh environmental conditions, and conduct research using more accurate models.  
\begin{table}
\begin{tabular}{l r r r r r} \hline \hline  & **Turning Range** & \(\mathbf{\delta}\) & **MIA** & **MAE** & **MTV** \\ \hline NF+CGSA & 2.23\(L\) & 10.16 & 12.08 & 36.02 & 0.14 \\ [PERSON] et al. [28] & 2.35\(L\) & 10.57 & 12.38 & 37.36 & 0.18 \\ PID & 2.58\(L\) & 10.66 & 12.30 & 39.14 & 0.15 \\ Decline ratio & 5.11\% & 3.88\% & 2.43\% & 3.60\% & 22.22\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of parameters of automatic U-turn in port at a water depth of 15 m.  
Figure 14: Ship trajectory under a Beaufort wind scale of No. 5, \(\psi_{T}=135^{\circ}\)[28].  
**Author Contributions:** Conceptualization, S.G. and X.Z.; methodology, S.G.; software, S.G.; validation, S.G. and X.Z.; formal analysis, S.G.; investigation, S.G.; resources, S.G.; data curation, S.G.; writing--original draft preparation, S.G.; writing--review and editing, S.G. and X.Z.; visualization, S.G.; supervision, X.Z.; project administration, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work is partially supported by the National Science Foundation of China (Grant No. 52171291), Dalian Innovation Team Support Plan in the Key Research Field (Grant No. 2020 RT08), Doctoral Research Initial Fund Project of Liaoning Province (No. 2021-BS-078), Dalian Maritime University University-level Team \"Open Ranking\" Project (No. 3132023502), the University 111 Project of China (Grant No. B08046), and International cooperation training program for innovative talents of Chinese Scholarships Council (Grant No. CSC [2022] 2260).  
**Data Availability Statement:** The data are contained within the article.  
**Acknowledgments:** The authors are grateful to each reviewer for their valuable comments and suggestions which improved the quality of this article. The authors would like to thank anonymous reviewers for their valuable comments which improved the quality of this article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.