## 2 Extension and Refinement of the Ocean Disturbance Simulation System Model
### Building the Ocean Current Disturbance Module  
To better illustrate the ocean currents, we decomposed the three-dimensional model, as shown in Figure 7.  
Based on Figure 7, if the angle between the current and the \(XOY\) plane was defined as \(\varphi\), the angle between the current and the \(ZOX\) plane was \(\theta\), and the speed of the ocean current was \(v_{c}\). According to the knowledge of trigonometric functions, we can obtain the formula for calculating the components of the ocean current in all directions as follows:  
\[V_{current}\left\{ \begin{aligned} V_{x}^{(1)}&=v_{c} \times cos\beta\ \times cos\alpha\\ V_{y}^{(1)}&=v_{c}\times sin\beta\ \times cos\alpha\\ V_{z}&=-v_{c}\times sin\alpha\end{aligned} \right. \tag{11}\]  
Wake flow happened in the horizontal direction, making it easier to analyze. Figure 8 shows the decomposed wake flow model. Figure 8: Wake flow example model. Figure 7: Ocean current example model. Using the same method as in Equation (10), we can obtain the formula for calculating the components of the wake flow in all directions as follows:  
\[V_{\text{wake}} \begin{cases}V_{x}^{(2)}=v_{f}\times cos\beta\\ V_{y}^{(2)}=v_{f}\times sin\beta\end{cases}  \tag{12}\]  
Thus, based on Figure 8 and Equations (8)-(11), we can calculate the influence of ocean currents and wake flow (\(v_{f}\)) as follows:  
* Non-wake flow area: \(V=V_{\text{current}}\);
* Wake flow area 1: \(V=V_{\text{current}}+V_{\text{wake}}\), that is: \[V_{\text{final}} \begin{cases}V_{x}=V_{x}^{(1)}+V_{x}^{(2)}\\ V_{y}=V_{y}^{(1)}+V_{y}^{(2)}\\ V_{z}=V_{z}\end{cases} \] (13)
* Wake flow area 2: \(V=V_{\text{current}}-V_{\text{wake}}\), which means: \[V_{\text{final}} \begin{cases}V_{x}=V_{x}^{(1)}-V_{x}^{(2)}\\ V_{y}=V_{y}^{(1)}-V_{y}^{(2)}\\ V_{z}=V_{z}\end{cases} \] (14)  
The ROV's approach direction relative to the ocean current was determined based on the initial and target positions, as illustrated in Figure 4. The approach angle \(\omega\), defined in Figure 6, was used to classify the flow interaction region and to select the appropriate control model accordingly:  
* Non-wake flow area (\(\omega\in[-45^{\circ},+45^{\circ}]\)): When the ROV moved approximately in the same direction as the ocean current, approaching from the front without entering the wake region, Equation (11) was applied;
* Wake flow area 1: (\(\omega\in(45^{\circ},\,135^{\circ})\cup(-135^{\circ},\,-45^{\circ})\)): When the ROV experienced lateral wake interference due to side-on current interaction, Equation (13) was used;
* Wake flow area 2: (\(\omega\in[135^{\circ},\,225^{\circ}]\)): When the ROV moved directly against the current, entering the downstream wake region behind the target, Equation (14) was employed. This classification balanced hydrodynamic realism with computational simplicity by adapting the model to varying current interaction scenarios. According to the previous analysis, to incorporate the influence of ocean currents on the underwater robot's trajectory, we only needed to modify the relative velocity components in the drag matrix's \(X\), \(Y\), and \(Z\) directions (shown in Figure 2). Specifically, the relative velocity was defined as the difference between the vehicle's body-fixed velocity and the ocean current velocity projected into the same frame. This approach followed the conventional assumption in underwater vehicle hydrodynamic modeling that the ocean current affects the system linearly by modifying the relative velocity vector, without introducing additional nonlinear terms [22; 23]. Thus, the nonlinear terms in the dynamic model remained unchanged. The updated relative velocity vector was then used to calculate hydrodynamic damping forces, and the modified drag matrix [24] was expressed as:  
\[\text{D}_{\text{current}}= \begin{bmatrix}D_{u1}&0&0&0&0&0\\ 0&D_{v1}&0&0&0&0\\ 0&0&D_{w1}&0&0&0\\ 0&0&0&K_{p}&0&0\\ 0&0&0&0&M_{q}&0\\ 0&0&0&0&0&N_{r}\end{bmatrix}