## 2 Dynamic Modeling of Trans-Medium Process
### Reference Frame  
In order to describe the motion of vehicles intuitively in three-dimensional space, two coordinate systems need to be defined as show in Figure 2 to establish a mathematical model. Since this study focuses on cross-domain phase motion, the \(z\)-axis direction in both the inertial coordinate system and the body-fixed coordinate system is defined as being upward. \(o_{p}x_{b}y_{p}z_{b}\) is the body-fixed coordinate system, whose origin is defined at the center of gravity of the HAUV. The inertial coordinate frame \(o_{e}x_{e}y_{e}z_{e}\) is an upper northeast coordinate system, and the \(x_{e}o_{e}y_{e}\) plane coincides with the static water surface. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable Symbol** & **Physical Meaning** & **Variable Symbol** & **Physical Meaning** \\ \hline \(o_{b}x_{b}y_{p}z_{b}\) & body-fixed coordinate system & \(V\) & volume of watertight chamber \\ \(o_{e}x_{e}y_{e}z_{e}\) & inertial coordinate frame & \(g\) & acceleration of gravity \\ \(\eta_{1}=\begin{bmatrix}x&y&z\end{bmatrix}^{\mathrm{T}}\) & position vector in inertial & \(\begin{bmatrix}x_{b}&y_{b}&z_{b}\end{bmatrix}^{\mathrm{T}}\) & center coordinate of buoyancy \\ \(\eta_{2}=\begin{bmatrix}\phi&\theta&\psi\end{bmatrix}^{\mathrm{T}}\) & attitude vector in inertial & \(M_{bw}\) & buoyancy torque \\ \(V_{1}=\begin{bmatrix}u&v&w\end{bmatrix}^{\mathrm{T}}\) & linear velocities in the body-fixed frame & \(\begin{bmatrix}T=\begin{bmatrix}T_{x}&T_{y}&T_{z}\end{bmatrix}^{T}\end{bmatrix}^{T}\) & control force \\ \(V_{2}=\begin{bmatrix}p&q&r\end{bmatrix}^{\mathrm{T}}\) & angular velocities in the body-fixed frame & \(\begin{bmatrix}R\big{(}s_{x},s_{y}\big{)}&\text{mapping transformation matrix}\\ \end{bmatrix}\) & \(\begin{bmatrix}\text{lift coefficients of the upper and}\\ k_{u},\,k_{\beta}&\text{lower propellers}\end{bmatrix}\) \\ \(I=\begin{bmatrix}I_{x}&I_{y}&I_{z}\end{bmatrix}\) & the moments of inertia & \(M_{\mathrm{T}}=\begin{bmatrix}u_{2}&u_{3}&u_{4}\end{bmatrix}^{\mathrm{T}}\) & control torque \\ \(m_{a}=diag\begin{bmatrix}m_{i}&m_{i}&m_{i}\end{bmatrix}\) & added mass & \(\begin{bmatrix}d\\ d\end{bmatrix}\) & lumped uncertainty and upper bound \\ \(I_{a}=\begin{bmatrix}I_{b}&I_{b}&I_{r}\end{bmatrix}\) & added moments of inertia & \(V=\begin{bmatrix}v_{x}&v_{y}&v_{z}\end{bmatrix}^{\mathrm{T}}\) & position virtual control \\ \(m_{w},I_{w}\) & total mass and inertia & \(x\) & state vector \\ \(k_{s},k_{sa}\) & linearization coefficient & \(u_{1}\) & total lift force \\ \(K_{dw},\,K_{da}\) & drag coefficient matrices in & \(\phi_{d}\) & expected roll \\ \(F_{bw}\) & buoyancy & \(\theta_{d}\) & expected pitch \\ \(\rho\) & density of water & \(\sigma\) & lift loss coefficient of the lower \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Related physical quantities and notation. Figure 2: The coaxial HAUV’s conceptual sketch and coordinate systems. The position and the attitude vector of the vehicle are defined as \(\mathbf{\eta}_{1}= \begin{bmatrix}x&y&z\end{bmatrix} ^{\text{T}}\) and \(\mathbf{\eta}_{2}= \begin{bmatrix}\phi&\theta&\psi\end{bmatrix} ^{\text{T}}\), respectively, in the earth-fixed reference frame. Furthermore, the horizontal motion of the HAUV is coupled to the roll and pitch, so the fuselage tilts during maneuvering, but it does not allow the fuselage to be flipped to avoid the singular problems of Euler angle modeling. \(\mathbf{V}_{1}= \begin{bmatrix}u&v&w\end{bmatrix} ^{\text{T}}\) and \(\mathbf{V}_{2}= \begin{bmatrix}p&q&r\end{bmatrix} ^{\text{T}}\) are defined as the linear and angular velocities, respectively, in the body-fixed frame. \[\mathbf{R}_{1}= \begin{bmatrix}c\theta\cdot c\psi&\text{s}\phi\cdot\text{s}\phi\cdot \text{c}\phi\cdot\text{s}\phi\cdot\text{c}\phi\cdot\text{s}\psi&c\phi\cdot \text{s}\theta\cdot c\psi+\text{s}\phi\psi\\ c\theta\cdot s\psi&\text{s}\phi\cdot\text{s}\theta\cdot\text{s}\psi+c\phi \cdot\text{c}\psi&c\phi\cdot\text{s}\theta\cdot s\psi-\text{s}\phi c\psi\\ -\text{s}\theta&\text{s}\phi\cdot c\theta&c\phi\cdot c\theta\end{bmatrix}