# Concise Robust Control of the Nonlinear Ship in Course-Keeping Control
## 3 Mathematical Model of Ship Motion  
The integrated modeling method of ship motion has a strict mathematical foundation. Because the interaction between various parts of the ship has been automatically taken into account in the hydrodynamic derivatives experiment, it has higher accuracy theoretically.  
From the point of view of the Abkowitz nonlinear ship motion model, the third-order dynamic nonlinear model of the ship is developed from the Taylor series. The general expression of the hydrodynamic force of ship motion is stated as follows:  
\[\left\{\begin{array}{l}m(\dot{u}-vr-x_{G}r^{2})=X(\dot{u},\dot{v},\dot{r},u, v,r,n,\delta)\\ m(\dot{v}+ur+x_{G}\dot{r})=Y(\dot{u},\dot{v},\dot{r},u,v,r,n,\delta)\\ I_{2z}\dot{r}+mx_{G}(\dot{v}+ur)=N(\dot{u},\dot{v},\dot{r},u,v,r,n,\delta)\end{array}\right. \tag{2}\]  
Among them, \(m\) is the mass of ship hull; \(\dot{u},\dot{v},\dot{r},u,v,r\) are the longitudinal acceleration, lateral acceleration, yawing angle acceleration, and the corresponding velocity components, respectively; \(n,\delta\) is the propeller revolution and rudder angle, respectively; \(x_{G}\) are the distance between the hull center and the centerline; \(X\), \(Y\) represent the longitudinal and lateral forces on the hull, respectively; \(N\) is the turning moment.  
The above formula is expanded by the Taylor series near the equilibrium point, and the higher-order terms above the third order are ignored in the control design. In addition, the terms of rigid inertia force, fluid inertia force acting on the hull, lift, and resistance terms are substituted into the above formula; hence, the non-dimensional Abkowitz nonlinear mathematical model of ship motion is obtained:  
\[\left\{\begin{array}{l}(m-X_{\dot{u}})\dot{u}=f_{1}(u,v,r,\delta)/0.5\rho L^{ 3}\\ (m-X_{\dot{v}})\dot{v}=f_{2}(u,v,r,\delta)/0.5\rho L^{3}\\ (mx_{G}-N_{\dot{v}})\dot{v}+(I_{zz}-N_{\dot{r}})\dot{r}=f_{3}(u,v,r,\delta)/0. 5\rho L^{4}\end{array}\right. \tag{3}\]  
where:  
\[\begin{array}{l}f_{1}(u,v,r,\delta)=X_{0}+X_{u}\Delta u+X_{uu}\Delta u^{2}+X _{uuu}\Delta u^{3}+X_{vv}\Delta v^{2}\\ \qquad+(X_{rr}+mx_{G})r^{2}+X_{\delta\delta}\delta^{2}+(X_{vr}+m)vr+X_{\delta v }\delta\\ \qquad+X_{\kappa\delta}\delta+X_{vvu}v^{2}\Delta u+X_{rr}r^{2}\Delta u\\ \qquad+X_{\delta\mu}\delta^{2}\Delta u+X_{v\mu}r\Delta u+X_{v\delta\
u}v \delta\Delta u+X_{r\delta u}r\delta\Delta u\\ f_{2}(u,v,r,\delta)=Y_{0}+Y_{u}\Delta u+Y_{uu}\Delta u^{2}+Y_{uuu}\Delta u^{3} +Y_{v}v+Y_{r}r+mru\\ \qquad+Y_{\delta}\delta+Y_{vv}\Delta v^{3}+Y_{rr}\Delta^{3}+Y_{\delta\delta \delta}\delta^{3}+Y_{vv}v^{2}r^{2}+Y_{v\delta v}v^{2}\delta\\ \qquad+Y_{\gamma r\
u}r^{2}+Y_{\delta r\delta}r^{2}+Y_{v\delta v}v\delta^{2}+Y _{r\delta\delta}r\delta^{2}+Y_{r\delta v}v\delta r\delta+Y_{uu}v\Delta u\\ \qquad+Y_{\gamma r\mu}\Delta u+Y_{\delta u}\delta\Delta u+Y_{\gamma u\
u}v \Delta u^{2}+Y_{ruru}\Delta u^{2}+Y_{\delta uu}\delta\Delta u^{2}\\ \qquad+N_{\delta}\delta+N_{vvv}\Delta v^{3}+N_{rrr}\Delta^{3}+N_{\delta\delta \delta}\delta^{3}\\ \qquad+N_{vvv}\
u^{2}r+N_{v\delta v}v^{2}\delta+N_{\gamma r\
u}r^{2}+N_{ \delta r\delta}r\delta r^{2}\\ \qquad+N_{v\delta}v\delta^{2}+N_{r\delta v}\delta^{2}+N_{r\delta v}v\delta+N_{ vu}v\Delta u\\ \qquad+N_{ru}r\Delta u-mx_{G}r\Delta u+N_{\delta u}\delta\Delta u+N_{vuu}v \Delta u^{2}\\ \qquad+N_{tuu}r\Delta u^{2}+N_{\delta uu}\delta\Delta u^{2}\end{array} \tag{5}\]  
Among them are the hydrodynamics derivatives except for \(u,v,r,\delta\), which could be determined by experiments. Among the parameters on the right side, \(X\), \(Y\), \(N\) are hydrodynamic derivatives.