# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 2 System Design and Implementation
The modification of the drag coefficient for the direct navigation of ships is shown as Equation (3):  
\[\frac{C_{th}}{C_{t\infty}}=A+\frac{B}{h/d} \tag{3}\]  
where \(C_{t\infty}\) is the ship's total drag coefficient in deep water; \(C_{th}\) is the ship's total drag coefficient at a water depth of \(h\); \(A\) and \(B\) are the experimental coefficients of the ship model. The modification of the viscous hydrodynamic derivatives is conducted as follows:  
\[\frac{X_{vrh}}{X_{\rm{proo}}}=1-0.9879\frac{d}{h}+21.9123\left(\frac{d}{h} \right)^{2}-73.8161\left(\frac{d}{h}\right)^{3}+71.1409\left(\frac{d}{h} \right)^{4} \tag{4}\]  
where \(X_{\rm{proo}}\) is the hydrodynamic derivative in deep water; \(X_{vrh}\) is the hydrodynamic derivative at a water depth of \(h\). The modification of viscous hydrodynamic derivatives is shown as in Equation (5):\[\left\{ \begin{array}{l}\frac{m_{x\infty}}{m_{\infty}}=\left[\left(\frac{h}{d}-1 \right)^{1.3}+3.77+1.14\frac{B}{d}-0.233\frac{L}{d}+3.43C_{b}\right]\left/ \left(\frac{h}{d}-1\right)^{1.3}\right.\\ \frac{m_{y\phi}}{m_{\psi\psi}}=\left[\left(\frac{h}{d}-1\right)^{0.82}+0.413+0.0 320\frac{B}{d}+0.0129\left(\frac{B}{d}\right)^{2}\right]^{2}\left/\left(\frac{ h}{d}-1\right)^{0.82}\right.\\ \left.\frac{l_{x\phi}}{l_{x\phi\psi}}=\left[\left(\frac{h}{d}-1\right)^{0.82}+0. 413+0.0192\frac{B}{d}+0.00554\left(\frac{B}{d}\right)^{2}\right]\left/\left( \frac{h}{d}-1\right)^{0.82}\right.\end{array} \right. \tag{5}\]  
where \(m_{x\infty}\), \(m_{y\infty}\), and \(J_{22\infty}\) are the additional mass and additional moment of inertia in deep waters. The modifications of the accompanying coefficient, hydrodynamic derivative to the propeller, and the rudder force are shown as Equations (6) and (7):  
\[\frac{\left(1-\omega_{p}\right)_{h}}{\left(1-\omega_{p}\right)_{\infty}}= \cos\left(1.4C_{b}\frac{d}{h}\right) \tag{6}\]  
\[\left\{ \begin{array}{l}\frac{\gamma_{Rb}}{\gamma_{R\infty}}=1+0.0161\frac{d }{h}+4.4222\left(\frac{d}{h}\right)^{2}-4.9825\left(\frac{d}{h}\right)^{3}\\ \frac{\alpha_{Hho}}{\alpha_{Hho}}=1+0.3621\frac{d}{h}+1.1724\left(\frac{d}{h} \right)^{2}\\ \frac{x_{Hho}}{\alpha_{Hho}}=1+0.3328\frac{d}{h}-3.2134\left(\frac{d}{h} \right)^{2}+2.5916\left(\frac{d}{h}\right)^{3}\end{array} \right. \tag{7}\]  
\(\omega_{p}\) is the hull accompaniment coefficient, \(\gamma_{R\infty}\) is the rectification coefficient of the hull and oars in deep water, \(\alpha_{H\infty}\) is the increase coefficient of the hull fluid force caused by steering, and \(x_{H\infty}\) is the central coordinate of the action center of the hull fluid force caused by steering. In this paper, different from the traditional ship model in the shallow water correction mentioned above, the third-order Nomoto model is converted into state-space form to facilitate the addition of environmental interference. Ship speed and water depth in the port and the ship draft are associated with the parameters of the ship motion model to describe the ship's low-speed movement in shallow water. Simulation experiments show that the model in this paper is simple and has certain accuracy. Ship motion in port needs to consider water depth, ship speed, the external nonlinear force, as well as the interference of wind and wave. By processing the parameters of the Nomoto model, the relationship between the model and water depth and ship speed can be obtained, while the state-space form of the second-order Nomoto model is inconvenient for adding the interference of the nonlinear force and external interference. Compared to higher-order and nonlinear models, the third-order model has a relatively simple structure, with parameters that are easy to identify and adjust, facilitating controller design and system analysis. At the same time, it can accurately reflect the motion characteristics of ships in shallow-water restricted areas, meeting the basic requirements for model accuracy in research.