## 2 Dynamic Modeling  
In this paper, the studied object is the coaxial HAUV, as shown in Figure 2. In order to accurately describe the motion of the vehicle and the dynamics model, two coordinate systems should be given first: the inertial coordinate system \(R_{E}\{O_{e}x_{e}y_{e}z_{e}\}\) and the body-fixed coordinate system \(R_{B}\{O_{b}x_{b}y_{b}z_{b}\}\) will be defined. The origin of the body-fixed coordinate system is set at the position of the heavy center of the vehicle. The inertial co-ordinate system is fixed at sea level, plane \(x_{e}O_{e}y_{e}\) coincides with a calm water surface, and \(O_{e}z_{e}\) points in the direction of the sky. The position and linear velocity vector of the HAUV are defined as \(\eta_{1}=\begin{bmatrix}x&y&z\end{bmatrix}^{\text{T}}\), \(\
u_{1}=\begin{bmatrix}u&v&w\end{bmatrix}^{\text{T}}\), and the attitude angle and angular velocity of the vehicle are denoted as \(\eta_{2}=\begin{bmatrix}\phi&\theta&\psi\end{bmatrix}^{\text{T}}\), \(\omega=\begin{bmatrix}p&q&r\end{bmatrix}^{\text{T}}\).  
Here, \(J\) is the transformation matrix from the body coordinate system to the inertial coordinate system.  
The kinematic equation of the coaxial HAUV is expressed as follows:  
\[\dot{\eta}=J\
u \tag{1}\]  
The meanings of each variable and symbol in the equation are as follows:  
\[\eta=\begin{bmatrix}\eta_{1}^{\text{T}}&\eta_{2}^{\text{T}}\end{bmatrix}^{ \text{T}},\
u=\begin{bmatrix}v_{1}^{\text{T}}&\omega^{\text{T}}\end{bmatrix}^{ \text{T}},J=\begin{bmatrix}J_{1}&0^{3\times 3}\\ 0^{3\times 3}&J_{2}\end{bmatrix},\]  
\[J_{1}=\begin{bmatrix}\cos\theta\cos\psi&\sin\phi\sin\theta\cos\psi-\cos\phi\sin \psi&\cos\phi\sin\theta\cos\psi+\sin\phi\text{sin}\psi\\ \cos\theta\sin\psi&\sin\phi\sin\theta\sin\psi+\cos\phi\cos\psi&\cos\phi\sin \theta\sin\psi-\sin\phi\cos\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\end{bmatrix},\]  
\[J_{2}=\begin{bmatrix}1&\sin\phi\tan\theta&\cos\phi\tan\theta\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi/\cos\theta&\cos\phi/\cos\theta\end{bmatrix}\]  
**Assumption 1**.: _To facilitate the controller design, we simplify the HAUV model and make the following basic assumption. The main mass of the HAUV is uniformly distributed on the cylindrically shaped watertight hull, and its geometric structure is symmetric about three coordinate planes. High-order hydrodynamic coefficients will not be considered._  
Since the main structure of the HAUV is uniform, it can be approximately considered that variables such as additional mass, buoyancy, and drag are linearly related to the height of the submerged part of the vehicle [23; 39]. Therefore, a switching coefficient is adopted to approximate the change trend of additional variables in the trans-medium process in the trans-medium stage. The existing switching coefficients mainly have a step form \(k_{sn}\)[14] and a linearized form \(k_{ln}\)[25; 39]. The two switching functions are discontinuous and easily  
Figure 2: Structure design and coordinate system of the coaxial HAUV.  
cause controller instability. This paper proposes a switching function, \(k_{th}\), based on the hyperbolic tangent function, which makes the model switching smoother and more stable.  
\[\begin{split} k_{tn}&=\left\{\begin{array}{c}0,z\geq h/2 \\ 1/2(1-2z/h),-h/2<z<h/2\\ 1,z\leq-h/2\end{array}\right.\\ k_{sn}&=1/2(1-\text{sign}(z))\\ k_{th}&=1/2(1-\text{tanh}(z/\mu))\end{split} \tag{2}\]  
where \(h\) is the altitude of the vehicle, \(z\) represents the height value of the vehicle in the inertial coordinate system, and \(0<\mu<1\) is the parameter to be designed. \(k_{s}=k_{tanh}\) represents the approximate change in the hydrodynamics in the transmedia process; \(k_{sa}=1-k_{s}\) indicates the switching coefficient of the aerodynamics, which describes the trend of the influencing factors in the air.  
Next, the cross-media dynamic evolution process of parameters and variables in the model will be analyzed.