# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
###### Abstract  
This study presents an innovative control strategy for enabling ships to perform automatic U-turns in restricted waters, with a focus on minimizing energy consumption and reducing wear on the steering gear. The strategy integrates a closed-loop gain-shaping algorithm with nonlinear feedback control, applied to a nonlinear motion mathematical model specifically designed for low-speed operations in shallow waters. The simulations, conducted under a Beaufort wind scale conditions up to No. 5 and water depths of 15 m, demonstrate that ships can successfully execute automatic U-turns within a distance three times their length. The incorporation of nonlinear feedback technology significantly reduces energy consumption and steering gear wear, with specific improvements including a reduction in the average rudder angle by up to 18.26%, a reduction in the mean absolute error (MAE) by up to 3.6%, a reduction in the mean integrated absolute (MIA) by up to 13.55%, and a reduction in the mean total variation (MTV) by up to 36.36%. These enhancements not only optimize the control effect but also extend the service life of the steering gear, thereby contributing to more sustainable maritime operations. Theoretical proofs and Matlab-based simulations validate the effectiveness of the controller, highlighting its potential for energy savings and improved navigational efficiency in challenging maritime environments. ship motion control; ship U-turn; restricted waters; closed-loop gain-shaping algorithm; nonlinear feedback technology +
Footnote †: journal: _Applied system_  
18 December 202418 January 2  
settings and with smaller vessels. However, they face significant limitations in complex port environments, especially with larger ships. Manual operations struggle to accurately control the ship's movement trajectory and are susceptible to environmental factors such as wind and currents, leading to prolonged turning times and heightened operational risks. With advancements in control theory, model-based control methods have been increasingly applied to ships, including sliding mode control [1,2], adaptive control [3,4], and fuzzy control [5,6]. For instance, [PERSON] et al. [7] explored the impact of water depth and speed on ship maneuverability and control and proposed an adaptive controller to enhance performance in shallow waters based on extensive experimental studies. The authors of [PERSON] et al. [8] investigated the Boussinesq-Burgers system for shallow water waves, identifying transformations and similarity reductions to aid in understanding wave behavior near ocean beaches, which has implications for ocean engineering fields. The authors of [PERSON] et al. [9] presented a modified shallow water equations model incorporating a scalar transport equation, utilizing a finite volume central-upwind scheme and an implicit-explicit Runge-Kutta method for the accurate and efficient simulation of geophysical flows with vertical profiles. The authors of [PERSON] and [PERSON] [10] examined the combined effects of various loading and wind conditions on a container ship's course-keeping performance, aiming to improve safe navigation practices by understanding the ship's response to different environmental and operational scenarios. The authors of [PERSON] et al. [11] introduced a particle swarm optimization (PSO)-based predictive PID-backstepping controller to enhance ship course-keeping performance under varying conditions, demonstrating improved efficiency and stability through simulation tests. The authors of [PERSON] et al. [12] utilized computational fluid dynamics simulations to analyze the maneuvering characteristics of a container ship in shallow waters, revealing significant changes in roll and turning diameter compared to deeper waters, with results that align well with experimental data. The authors of [PERSON] et al. [13] presented an integrated framework using enhanced LQR control and adaptive strategies for autonomous ship berthing, validated through simulations under various operating conditions.