## 2 Extension and Refinement of the Ocean Disturbance Simulation System Model
### Theoretical Basis
The coordinate system used for defining these quantities is illustrated in Figure 2. \(\mathbf{M}_{\mathbf{A}}\) is the additional mass term; and \(\mathbf{X}_{ii}\), \(\mathbf{Y}_{\psi}\) and \(\mathbf{N}_{\mathbf{r}}\) are the added mass coefficients on the three directions of freedom, respectively. The terms such as \(Z_{g}\), \(Z_{w}\), and similar off-diagonal components reflect the coupling effects arising from the offset between the center of gravity (CoG) and the center of buoyancy (CoB). Specifically, (\(x_{g}\), \(y_{g}\), \(z_{g}\)) represents the coordinates of the CoG, which is the point where the total gravitational force is considered to act. These coordinates are determined by the mass distribution of the vehicle. (\(x_{b}\), \(y_{b}\), \(z_{b}\)) denotes the coordinates of the CoB, which is the centroid of the displaced fluid volume and the point where the buoyant force acts. These coordinates depend on the geometry and submerged volume of the vehicle. Therefore, **M** can be expressed as follows: \[\text{M}= \begin{bmatrix}m-X_{\hat{u}}&0&0&0&mz_{g}&-my_{g}\\ 0&m-Y_{\hat{v}}&0&-mz_{g}&0&mx_{g}\\ 0&0&m-Z_{\hat{w}}&my_{g}&-mx_{g}&0\\ 0&-mz_{g}&my_{g}&I_{x}-K_{\hat{p}}&0&0\\ mz_{g}&0&-mx_{g}&0&I_{y}-M_{\hat{q}}&0\\ -my_{g}&mx_{g}&0&0&0&I_{z}-N_{\hat{r}}\end{bmatrix} ,\] (2)
* Coriolis moment matrix **C(v)** \[\text{C}(\
u)= \begin{bmatrix}0&0&0&0&mw&-mv\\ 0&0&0&-mw&0&mu\\ 0&0&0&mv&-mu&0\\ 0&mw&-mv&0&-rI_{z}&qI_{y}\\ -mw&0&mu&rI_{z}&0&-pI_{x}\\ mv&-mu&0&-qI_{y}&pI_{x}&0\end{bmatrix} ,\] (3) This matrix reflects the inertial coupling effect, mainly affecting the system's stability under high-speed motion [15; 16]. * Damping matrix **D(v)** The matrix reflects the damping effect of the surrounding water on the vehicle's motion and is divided into linear and nonlinear components. The linear damping model accounts for viscous resistance proportional to velocity, while the nonlinear damping accounts for drag forces that increase nonlinearly with speed, typically using quadratic terms. To simplify the hydrodynamic model and reduce computational complexity, the damping matrix is assumed to be diagonal, neglecting cross-coupling terms between different degrees of freedom--an approach commonly used for symmetric underwater vehicles operating at low to moderate speeds [17]. The nonlinear damping terms are represented in a simplified quadratic form, \(\propto\!|\,v\!|\,v\), which captures the dominant drag behavior in most practical scenarios and maintains computational efficiency. It is acknowledged, however, that more rigorous models may employ third-order terms (e.g., \(|\,v\!|\,v^{2}\)) or sign-varying cubic expressions to reflect direction-dependent energy dissipation, since nonlinear damping forces are not always strictly positive. Such complexities become important under oscillatory flow or dynamic maneu  
Figure 2: The coordinate system. vering. The current model adopts the simplified form as a practical trade-off between model fidelity and computational tractability:  
\[\mathbf{D_{lin}}=\text{diag}(X_{u},Y_{v},Z_{uv},K_{p},M_{q},N_{r}), \tag{4}\]  
\[\mathbf{D_{nonlin}}=\text{diag}(X_{|u|}|u|,Y_{|v|}|v|Z_{|u|}|w|_{K|p|}|p|M_{|q| }|q|,N_{|r|}|r|), \tag{5}\]  
\[\mathbf{D(v)}=\mathbf{D_{lin}}+\mathbf{D_{nonlin}}, \tag{6}\]  
Note that although the damping matrix \(\mathbf{D(v)}\) includes terms proportional to the absolute value of the velocity, all damping coefficients (e.g., \(Xu,X|u|\)) are strictly non-negative. When the damping force is computed as \(-\mathbf{D(v)v}\), the result always opposes the direction of motion, ensuring that the damping effect is dissipative. Therefore, \(\mathbf{D(v)}\) is positive definite (or semi-definite) in the sense of energy dissipation and does not introduce any energy into the system. The parameters can be obtained by CFD simulation or experimental calibration [18]. * Gravity and buoyancy matrix \(\mathbf{G(\eta)}\) The matrix reflects the restoring force caused by gravity and buoyancy under attitude change.