## 2 Methodology
### Equation of Motion of the Submarine  
Considering the motion of the submarine with six degrees of freedom, the equations for the submarine's motion in calm water in the body-fixed coordinate system have been described by [PERSON] (2014), as follows:  
\[m\Big{[}\dot{u}-vr+wq-x_{\text{G}}\Big{(}q^{2}+r^{2}\Big{)}+y_{\text{G}}\big{(}pq -\dot{r}\big{)}+z_{\text{G}}\big{(}pr+\dot{q}\big{)}\Big{]}=X \tag{1}\]  
\[m\Big{[}\dot{v}-wp+ur-y_{\text{G}}\Big{(}r^{2}+p^{2}\Big{)}+z_{\text{G}}\big{(} qr-\dot{p}\big{)}+x_{\text{G}}\big{(}qp+\dot{r}\big{)}\Big{]}=Y \tag{2}\]  
\[m\Big{[}\dot{w}-uq+vp-z_{\text{G}}\Big{(}p^{2}+q^{2}\Big{)}+x_{\text{G}}\big{(} rp-\dot{q}\big{)}+y_{\text{G}}\big{(}rq-\dot{p}\big{)}\Big{]}=Z \tag{3}\]  
\[I_{x}\dot{p}+\big{(}I_{z}-I_{y}\big{)}qr-I_{xz}\big{(}\dot{r}+pq\big{)}+I_{yz} \big{(}r^{2}-q^{2}\big{)}+I_{xy}\big{(}qp-\dot{q}\big{)} \tag{4}\]  
\[+m\big{[}y_{\text{G}}\big{(}\dot{w}-uq+vp\big{)}-z_{\text{G}}\big{(}\dot{v}-wp +ur\big{)}\big{]}=K\]  
\[I_{y}\dot{q}+\big{(}I_{x}-I_{z}\big{)}rp-I_{xy}\big{(}\dot{p}+qr\big{)}+I_{zx} \big{(}q^{2}-p^{2}\big{)}+I_{yz}\big{(}qp-\dot{r}\big{)}\]  
\[+m\big{[}z_{\text{G}}\big{(}\dot{u}-vr+wq\big{)}-x_{\text{G}}\big{(}\dot{w}-uq +vp\big{)}\big{]}=M \tag{5}\]  
\[I_{z}\dot{r}+\big{(}I_{y}-I_{x}\big{)}pq-I_{yz}\big{(}\dot{q}+rp\big{)}+I_{xy} \big{(}q^{2}-p^{2}\big{)}+I_{zx}\big{(}rq-\dot{p}\big{)}\]  
\[+m\big{[}x_{\text{G}}\big{(}\dot{v}-wp+ur\big{)}-y_{\text{G}}\big{(}\dot{u}-vr+ wq\big{)}\big{]}=N \tag{6}\]  
where \(x_{\text{G}}\), \(y_{\text{G}}\), and \(z_{\text{G}}\) indicate the coordinate of center of gravity along the \(x\)-, \(y\)-, and \(z\)-axes, respectively; \(I_{x}\), \(I_{y}\), and \(I_{z}\) are mass moments of inertia of the ship about the \(x\)-, \(y\)-, and \(z\)-axes, respectively; \(I_{xy}\), \(I_{yz}\), and \(I_{zx}\) are the products of inertia about the \(xy\)-, \(yz\)- and \(zx\)-axes, respectively; and \(m\) is the mass of the ship.  
Regarding the submarine's motion in calm water, the external force and moment in the right side of the equations can be divided into hull forces (\(F_{H}\)), propeller forces (\(F_{P}\)), and control forces (\(F_{C}\)). To steer the submarine with six degrees of freedom, the control force and moment of the rudders and stabilizers should be determined. These are then converted to a corresponding dimensionless form through dividing by (\(\dot{\
u}\dot{\
u}\)2V2L2) and (\(\dot{\
u}\dot{\
u}\)2V2L3), where L and V are the overall length and speed of the submarine, respectively, and \(\rho\) is the density of water. The hydrodynamic coefficients of the control plane are then obtained from the Taylor expansion, as shown in following equations  
\[F(X,Y,Z,K,M,N)=F_{H}+F_{P}+F_{\text{C}} \tag{7}\]  
\[X_{\text{C}}^{\prime}=X_{\delta_{x}\delta_{y}}^{\prime}\delta_{r}^{2}+X_{ \delta_{x}\delta_{y}}^{\prime}\delta_{s}^{2} \tag{8}\]  
\[Y_{\text{C}}^{\prime}=Y_{\delta_{x}}^{\prime}\delta_{r}+Y_{\delta_{x}|\delta_ {y}|}^{\prime}\delta_{r}|\delta_{r}| \tag{9}\]  
\[Z_{\text{C}}^{\prime}=Z_{\delta_{x}}^{\prime}\delta_{s}+Z_{\delta_{y}|\delta_ {s}|}^{\prime}\delta_{s}|\delta_{s}| \tag{10}\]  
\[K_{\text{C}}^{\prime}=K_{\delta_{x}}^{\prime}\delta_{r}+K_{\delta_{x}|\delta_ {r}|}^{\prime}\delta_{r}|\delta_{r}|\[N_{\text{C}}^{\prime}=N_{\delta_{j}}^{\prime}\delta_{r}+N_{\delta_{j,k_{i}}|\delta _{r}|}^{\prime}\delta_{r}|\delta_{r}| \tag{13}\]  
where the symbols on the right side of the equations are the hydrodynamic coefficients of the stern rudder plane configurations.