## 2 Dynamic Modeling
### Analysis of the Interference Mathematical Model
#### 2.6.3 Wave Interference  
In order to simplify the problem, wave interference caused by single-frequency micro-amplitude-long peak waves with a fixed direction is considered [40,44]. It is assumed that these waves are regular waves concentrated in the x-axis and y-axis directions.  
The corresponding velocity potential functions \(\Gamma_{x}\) and \(\Gamma_{y}\) re expressed as  
\[\begin{array}{l}\Gamma_{x}=\sum\limits_{i=1}^{N}\frac{gA_{xi}}{\omega_{xi}} e^{k_{xi}z}\sin(k_{xi}x-\omega_{xi}t+\varphi_{xi})\\ \Gamma_{y}=\sum\limits_{i=1}^{N}\frac{gA_{yi}}{\omega_{yi}}e^{ky^{2}}\sin(k_{ yi}y-\omega_{yi}t+\varphi_{yi})\end{array} \tag{19}\]  
where \(N\), \(A_{ji}\), \(\omega_{ji}\), \(k_{ji}\), and \(\varphi_{ji}(j=x,y)\) are the total number of waves, wave amplitude, wave frequency, wave number, and wave time, respectively. A simple dispersion relation is satisfied between the wave number and the circular frequency of a wave, i.e., \(k_{ji}=w_{ji}^{2}/g\).  
The transverse and vertical velocities of the wave in the x- and y-axis directions and the corresponding accelerations are given by the following equation:  
\[\begin{array}{l}u_{x}=\frac{\partial\Gamma_{x}}{\partial x},w_{x}=\frac{ \partial\Gamma_{x}}{\partial z},u_{y}=\frac{\partial\Gamma_{y}}{\partial y},w_ {y}=\frac{\partial\Gamma_{y}}{\partial z}\\ \dot{u}_{x}=\frac{\partial u_{x}}{\partial y},\dot{w}_{x}=\frac{\partial u_{x}} {\partial t},\dot{u}_{y}=\frac{\partial u_{y}}{\partial y},\dot{w}_{y}=\frac {\partial v_{y}}{\partial t}\end{array} \tag{20}\]The effect of the HAUV on the incident wave pressure field can be neglected. Therefore, the wave loads encountered by the vehicle at the water surface can be calculated according to the following equation [24]  
\[\begin{array}{l}F_{wave,x}=\int_{L_{z}}(\rho_{w}C_{m}\pi R^{2}\dot{u}_{x}+\frac {1}{2}\rho_{w}C_{d}Ru_{x}|u_{x}|)dz\\ F_{wave,y}=\int_{L_{z}}(\rho_{w}C_{m}\pi R^{2}\dot{u}_{y}+\frac{1}{2}\rho_{w}C_ {d}Ru_{y}|u_{y}|)dz\\ M_{wave}=r_{b}\times F_{wave}\\ F_{wave}=\int_{1}^{T}\left[F_{wave,x},F_{wave,y},0\right]^{T}\end{array} \tag{21}\]  
where \(R\) is the body radius of the coaxial HAUV and \(C_{m}\) and \(C_{d}\) are the inertia and drag coefficients of the watertight hull of the coaxial HAUV, respectively. \(L_{z}\) denotes the integral length of the cabin subjected to wave action, which is expressed as  
\[L_{z}\in\left\{\begin{array}{l}[0,0],z\geq H/2\\ [z-H/2,0],-H/2<z<H/2\\ [z-H/2,z+H/2],z\leq H/2\end{array}\right. \tag{22}\]  
where \(H\) is the body height of the coaxial HAUV, and \(z\) is the value of the altitude in the inertial frame.  
The above marine environmental parameters applied in the simulation are detailed in Appendix A, Table A1.  
**Remark 4**.: _Based on the detailed analysis of the various external forces and moments applied to the system in the previous subsection, they can be classified. The first category is the active control forces that need to be regulated by the controller, i.e., control forces and control moments, \(\tau_{0}\); the second category consists of external forces that can be measured or approximated, such as the Coriolis forces matrix, \(C_{v}\), restoring forces, \(g_{v}\), damping matrix., etc.; and the third category consists of terms that are not practically measurable or are extremely expensive to model accurately, such as wind-disturbing forces (\(F_{wind}\), \(M_{wind}\)), wave-disturbing forces (\(F_{wave}\), \(M_{wave}\)), current-disturbing forces (\(F_{c}\), \(M_{c}\)), and so on. The third type of external forces can be approximated using empirical equations, but it is very difficult to obtain real-time information about the marine environment, which has limited accuracy._