# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 2 System Design and Implementation  
_Mathematical Model of Ship Motion with Low Speed in Shallow Water_  
The motion of a ship in a port involves a movement with low speed and large drift angle, and the disturbance of the wind and waves in the port is relatively increased; the water depth is shallower than that in the general sea, which will lead to an increase in the ship's resistance and the deterioration of the rudder response. This will lead to some difficulties being encountered in controlling the ship. Since ships are divided into specifications and sizes, they need to be judged according to the ratio of the water depth to the ship's draft, \(h/d\). At present, there is no clear and unified concept definition for shallow water in the international community. For normal transport ships, water depth is usually divided into four ranges of standard: deep water (\(h/d>3.0\)), medium water depth (\(1.5<h/d<3.0\)), shallow water (\(1.2<h/d<1.5\)), and very shallow water (\(h/d<1.2\)). Generally speaking, when the ratio of water depth, \(h\), to draft, \(d\): \(h/d\), is less than 3, the shallow-water effect will occur. In shallow water, the hydrodynamic force and hydrodynamic torque of the ship increase due to an increase in the speed of water around the hull, and the main effect is reflected in the increase in the difficulty of berthing and turning around in port [19]. The automatic U-turn operation of ships is generally only applicable for medium- and small-sized ships under good environmental conditions and wide water areas. For large ships, the automatic U-turn operation is difficult to achieve, and tugs are usually used to assist the turning operation. The flow scene caused by the movement of ships in shallow waters is significantly different from that in deep waters. Generally speaking, the inertial fluid power acting on the hull--that is, the additional mass and additional moment of inertia of the hull--will increase with the decrease in water depth; the viscous fluid power acting on the hull also has the same situation. The coordinate point is taken at the center of the ship, and the ship motion model is shown as Equation (1):  
\[\left\{ \begin{array}{l}(m+m_{x})\dot{u}-\left(m+m_{y}\right)vr=X_{H}+X_{P}+X_ {R}\\ (m+m_{y})\dot{v}+(m+m_{x})ur=Y_{H}+Y_{P}+Y_{R}\\ (I_{zz}+I_{zz})\dot{r}=N_{H}+N_{P}+N_{R}-Y_{H}\cdot x_{c}\\ \dot{\psi}=r\\ x=u\cos\psi-\upsilon\sin\psi\\ y=v\cos\psi+u\sin\psi\end{array} \right. \tag{1}\]  
where \(m\) is ship mass; \(I_{zz}\) is moment of inertia of \(m_{x}\) and \(m_{y}\); \(I_{zz}\) is the additional mass and additional moment of inertia of the ship. \(X,Y\), and \(N\) represent the force and moment acting on the ship; the subscripts \(H,P\), and \(R\) represent the hull, propeller, and rudder, respectively; \(x_{c}\) represents the distance from the coordinate origin to the ship's center. \(x,y\) are the ship's position coordinates; \(\psi\) is the ship's heading angle; \(u,v\), and \(r\) represent the ship's surge velocity, sway velocity, and yaw rate, respectively. According to the Inoue model, the force and moment in Equation (1) can be calculated as Equation (2):  
\[\left\{ \begin{array}{l}X_{H}=X(u)+X_{vv}v^{2}+X_{vr}vr+X_{rr}r^{2}\\ Y_{H}=Y_{v}v+Y_{r}r+Y_{|v|v|}|v|+Y_{|r|r|}|r|+Y_{|v|r|}|v|\\ N_{H}=N_{v}v+N_{r}r+N_{|v|v|}|v|+N_{|r|r|}r|r+N_{vv}v^{2}r+N_{vr}vrr^{2}\end{array} \right. \tag{2}\]  
where \(X(u)\) represent the ship's direct sailing resistance; \(X(u)=-\frac{S}{L_{H}}C_{t}u^{2}\); \(S\) is the wet area of the hull; \(C_{t}\) is the ship's total drag coefficient; the coefficients preceding the remaining velocity terms are all hydrodynamic derivatives, which are related to the ship's own parameters. For specific calculations of hydrodynamic derivatives, see the Appendix A. In shallow-water conditions, the traditional ship model needs to be modified; the formulas are shown below.