## 3 Mathematical Model
### Six-Degree-of-Freedom Motion Equations  
According to [PERSON]'s Second Law, the equations of motion [7] to represent hydrodynamic forces and moments as functions of motion variables were introduced in Equations (15)-(20). The left-hand side of the equations is the rigid body dynamics, whereas the right-hand side represents the hydrodynamic force acting on the SFRM. \(1\)Surge  
\(m\left[\dot{u}-vr+wq-x_{G}\left(q^{2}+r^{2}\right)+y_{G}\left(pq-\dot{r}\right)+ z_{G}\left(pr+\dot{q}\right)\right]\)  
\(=\frac{\rho}{2}L^{4}\left[X_{qq}^{\prime}q^{2}+X_{rr}^{\prime}r^{2}+X_{rr}^{ \prime}rprp\right]+\frac{\rho}{2}L^{3}\left[X_{ii}^{\prime}\dot{u}+X_{rr}^{ \prime}vr+X_{qq}^{\prime}vqq\right]\)  
\(+\frac{\rho}{2}L^{2}\left[X_{uu}^{\prime}u^{2}+X_{pp}^{\prime}v^{2}+X_{ww}^{ \prime}w^{2}\right]+\frac{\rho}{2}L^{2}u^{2}\left[X_{\delta_{x}\delta_{y}}^{ \prime}{\delta_{r}}^{2}+X_{\delta_{x}\delta_{y}}^{\prime}{\delta_{z}}^{2}+X_{ \delta_{y}\delta_{x}}^{\prime}{\delta_{y}}^{\prime}{\delta_{y}}^{2}\right]\)  
\(-(W-B)\sin\theta+X_{P}-X_{R}\)  
Figure 3: (**a**) The earth-fixed coordinate system; (**b**) rear, (**c**) side, and (**d**) top views of the body-fixed coordinate system. 2 Sway \[ \begin{array}{ll}&m\big{[}\dot{v}+ur-wp+x_{G}\big{(}pq+\dot{r}\big{)}-y_{G} \big{(}p^{2}+r^{2}\big{)}+z_{G}\big{(}qr-\dot{p}\big{)}\big{]}\\ &=\frac{\varrho}{2}L^{4}\Big{[}Y^{\prime}_{p}\dot{p}+Y^{\prime}_{r}\dot{r}+Y^{ \prime}_{pq}pq+Y^{\prime}_{\dot{p}|p}|p|\big{]}\\ &+\frac{\varrho}{2}L^{3}\Big{[}Y^{\prime}_{p}\dot{v}+Y^{\prime}_{wp}wp+Y^{ \prime}_{v|r}|\frac{p}{|p|}\Big{|}\big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}}\Big{|} \big{|}|r+Y_{p}up+Y_{r}ur\Big{]}\\ &+\frac{\varrho}{2}L^{2}\Big{[}Y^{\prime}_{uu}u^{2}+Y_{p}uv+Y^{\prime}_{v| \varphi|}v\Big{]}\Big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}}\Big{|}+Y_{wp}ww\Big{]} +\frac{\varrho}{2}L^{2}u^{2}Y^{\prime}_{\delta_{r}}\delta_{r}\\ &+(W-B)\cos\theta\sin\phi+\frac{\varrho}{2}L^{2}\big{(}F_{y}\big{)}_{vs}\frac{ v^{2}+w^{2}}{u^{2}}(-w)\sin\omega t+WF_{s}\end{array} \] (16)
3 Heave \[ \begin{array}{ll}&m\big{[}\dot{w}-uq+vp+x_{G}\big{(}pr-\dot{q}\big{)}+y_{G} \big{(}qr+\dot{p}\big{)}-z_{G}\big{(}p^{2}+q^{2}\big{)}\big{]}\\ &=\frac{\varrho}{2}L^{4}\Big{[}Z^{\prime}_{q}\dot{q}+Z^{\prime}_{rr}r^{2}+Z^{ \prime}_{rp}rp\Big{]}+\frac{\varrho}{2}L^{3}\Big{[}Z^{\prime}_{\dot{w}}\dot{w} +Z^{\prime}_{vr}vr+Z^{\prime}_{rp}vp\Big{]}\\ &+\frac{\varrho}{2}L^{3}\Big{[}Z^{\prime}_{q}uq+Z^{\prime}_{v|q|}\frac{w}{|w| }\Big{|}\big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}}\Big{|}|q|\Big{]}\\ &+\frac{\varrho}{2}L^{2}\Big{[}Z^{\prime}_{uu}u^{2}+Z^{\prime}_{up}uw+Z^{ \prime}_{v|w|}w\Big{|}\big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}}\Big{|}\Big{]}\\ &+\frac{\varrho}{2}L^{2}\Big{[}Z^{\prime}_{|w|}u|w|+Z^{\prime}_{vww}\Big{]} \Big{(}w\big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}}+Z^{\prime}_{vv}v^{2}\Big{]}+ \frac{\varrho}{2}L^{2}u^{2}\Big{[}Z^{\prime}_{\delta_{r}}\delta_{s}+Z^{\prime }_{\delta_{b}}\delta_{b}\Big{]}\\ &+(W-B)\cos\theta\cos\phi+\frac{\varrho}{2}L^{2}(F_{z})_{vs}\frac{v^{2}+w^{2 }}{u^{2}}v\sin\omega t+WF_{H}\end{array} \] (17) Roll \[ \begin{array}{ll}&I_{x}\dot{p}+\big{(}I_{z}-I_{y}\big{)}qr-\big{(} \dot{r}+pq\big{)}I_{xz}+m\big{[}y_{G}\big{(}\dot{w}-uq+vp\big{)}-z_{G}\big{(} \dot{v}+ur-wp\big{)}\big{]}\\ &=\frac{\varrho}{2}L^{5}\Big{[}K^{\prime}_{\dot{p}}\dot{p}+K^{\prime}_{r}\dot {r}+K^{\prime}_{qr}qr+K^{\prime}_{p|p|}p|\big{]}\\ &+\frac{\varrho}{2}L^{4}\Big{[}K^{\prime}_{p}up+K^{\prime}_{r}ur+K^{\prime}_{ \dot{v}}\dot{v}+K^{\prime}_{wp}wp\Big{]}\\ &+\frac{\varrho}{2}L^{3}\Big{[}K^{\prime}_{uu}u^{2}+K^{\prime}_{v}uv+K^{ \prime}_{vw}w+K^{\prime}_{v|v|}\Big{]}\Big{(}v^{2}+w^{2}\big{)}^{\frac{1}{2}} \Big{|}\Big{]}+\frac{\varrho}{2}L^{3}u^{2}K^{\prime}_{\delta_{r}}\delta_{r}\\ &+(Wy_{G}-By_{B})\cos\theta\cos\phi-(Wz_{G}-Bz_{B})\cos\theta\sin\phi+WF_{R} \end{array} \] (18) Pitch \[