## 2 System Description and Analysis
### ACV Model
#### 2.2.4 Roll Restoring Moment  
The roll restoring moment generated by gravity and buoyancy is given as:  
\[K_{res}=-k_{res}mg\sin(\varphi)\approx-k_{res}mg\varphi \tag{15}\]where \(k_{res}\) is the roll restoring coefficient. Combining the above analysis with the kinematic equation of ACV [13], four DOF vector model of ACV is proposed as:  
\[\mathbf{V^{b}_{w}}=\mathbf{R}(\varphi,\psi)\mathbf{V^{e}_{w}} \tag{16}\]  
\[\hat{\mathbf{\eta}}=\mathbf{R}(\varphi,\psi)\mathbf{\
u} \tag{17}\]  
\[\mathbf{M\dot{v}}+\mathbf{C(v)\
u}+\mathbf{D(v)\
u}+\mathbf{E(V^{b}_{w},\eta)\
u}= \tag{18}\]  
where \(\mathbf{V^{b}_{w}}=[u^{b}_{w},v^{b}_{w},0,0]^{T}\) and \(\mathbf{V^{e}_{w}}=[u^{e}_{w},v^{e}_{w},0,0]^{T}\) represent the wind speed vectors in the body-fixed and earth-fixed frames, respectively. \(\mathbf{\eta}=[x,y,\varphi,\psi]^{T}\) is the position vector in the earth-fixed frame, consisting of the surge position \(x\), the sway position \(y\), the roll angle \(\varphi\in[-0.5\pi,0.5\pi]\), and the yaw angle \(\psi\in[0,2\pi]\), where \(\mathbf{\
u}=[u,v,p,r]^{T}\) is the velocity vector in the body-fixed frame, consisting of the surge velocity \(u\), the sway velocity \(v\), the roll velocity \(p\), and the yaw velocity \(r\) of the ACV. \(\mathbf{\tau}=[\tau_{u},0,0,\tau_{r}]^{T}\) is the input vector generated by the air propellers and air rudders, and \(\mathbf{W}=[w_{u},w_{v},w_{p},w_{r}]^{T}\) is the external disturbance vector in four DOFs. \(\mathbf{R_{res}(\eta)}=[0,0,K_{res},0]^{T}\) is the restoring moment. \(\mathbf{R}(\varphi,\psi)\) denotes the rotation matrix; \(\mathbf{M}\) denotes the inertia matrix; \(\mathbf{C(v)}\) denotes the skew-symmetric Coriolis; \(\mathbf{D(v)}\) denotes the resistance matrix; and \(\mathbf{E(V^{b}_{w},\eta)}\), which is the main difference compared with the traditional marine surface vessels, is also a kind of resistance matrix. They are given as follows:  
\[\mathbf{R}(\varphi,\psi)=\left[ \begin{array}{cccc}\cos(\psi)&-\sin(\psi)\cos( \varphi)&0&0\\ \sin(\psi)&\cos(\psi)\cos(\varphi)&0&0\\ 0&0&1&0\\ 0&0&0&\cos(\varphi)\end{array} \right]\]  
\[\mathbf{D(v)}=\mathbf{D_{h}}+\mathbf{D_{nh}(v)}+\mathbf{D_{a}(v)}=\left[ \begin{array}{cccc} d_{11}&d_{12}&0&0\\ d_{21}&d_{22}&0&d_{24}\\ d_{31}&d_{32}&0&d_{34}\\ d_{41}&d_{42}&0&d_{44}\end{array} \right]\]  
\[\mathbf{E(v)}=\left[ \begin{array}{cccc}k_{axu}&k_{axv}&0&0\\ k_{ayu}&k_{ayv}&0&0\\ k_{apu}&k_{ayv}&0&0\\ k_{amu}&k_{anv}&0&0\end{array} \right]\left[ \begin{array}{cccc}u^{b}_{w}\\ v^{b}_{w}\\ 0\\ 0\end{array} \right]\]  
\[\mathbf{W}=\left[ \begin{array}{cccc}k_{mxu}&0&0&0\\ 0&k_{myv}&0&0\\ 0&k_{mpv}&0&0\\ 0&k_{mpv}&0&0\\ k_{mmu}&k_{mmv}&0&0\end{array} \right]\left[ \begin{array}{cccc}u^{b}_{w}\\ v^{b}_{w}\\ 0\\ 0\end{array} \right]+\left[ \begin{array}{cccc}k_{axu}\\ k_{ayu}\\ k_{amu}\\ k_{amu}\end{array} \right](V^{e}_{w})^{2}\]  
\[ \begin{array}{ll}d_{11}=(\mu_{x}+k_{axu})u+k_{mxu},d_{12}=(\mu_{x}+k_{axv})v,d_{21}=(\mu_{y}+k_{axu})u\\ d_{22}=(\mu_{y}+k_{ayv})v+k_{myv},d_{24}=Y_{r}+Y_{|v|r}\,|v|+Y_{|r|}\,|r|\,d_{ 31}=(\mu_{k}+k_{aku})u\\ d_{32}=(\mu_{k}+k_{akv})v+k_{mkv},d_{34}=K_{r}+K_{|v|r}\,|v|+K_{|r|}\,|r|\\ d_{41}=(\mu_{n}+k_{amu})u+k_{mmu},d_{42}=(\mu_{n}+k_{anv})v+k_{mmv},d_{44}=N_{r }+N_{|v|r}\,|v|+N_{|r|}\,|r|\\ \end{array} \]  
Since \(\mathbf{E(V^{b}_{w},\eta)}\) and \(\mathbf{W(V^{b}_{w},\eta)}\) are affected by uncertain external disturbance, based on Lemma 1, the RBFNN can be used to approximate the unknown term \(\mathbf{W(V_{w}^{b},\mathbf{\eta})}\), define \(\mathbf{\bar{x}}=[\mathbf{\eta}^{T},\mathbf{v}^{T},\mathbf{V_{w}^{b}}]^{T}\) and \(\widehat{f}(\mathbf{x})=[\widehat{f}_{x}(\mathbf{x}),\widehat{f}_{y}(\mathbf{x}),\widehat{f} _{p}(\mathbf{x}),\widehat{f}_{n}(\mathbf{x})]^{T}\) as the input and output, respectively. Then \(\mathbf{f(\eta,\
u,V_{w}^{b})}=\mathbf{W^{T}H+\varepsilon}\). **Remark 1**: _Compared with the ACV model proposed in the previous literature [13], the vector model proposed above not only highlights the characteristics of the ACV but also simplifies the workload of the RBFNN.