## 3 Modeling of Path Planning
### Mathematical Definition of the Problem  
The sailing of the AUV is easily affected by ocean currents due to its small inertia. On the one hand, the ocean current will interfere with the navigation of AUVs, causing its navigation drift. On the other hand, the rational use of ocean currents can assist AUVs to reduce energy consumption. Considering two extreme cases, if the AUV's navigation direction is consistent with the ocean current direction, the ocean currents will provide the positive work for the AUV's navigation. In this case, the ocean current assists the AUV to sail. When the AUV's navigation direction is opposite to the ocean current, in such case the ocean currents do negative work to the AUV, and the currents hinder the AUV's sailing. Therefore, if the ocean current information of the known environment is considered in the local path planning stage, the influence of ocean currents on the AUV navigation can be utilized to some extent to assist the AUV navigation. The ocean current information is modeled as a two-dimensional ocean map, which represents the seawater flow information. This simplification is reasonable because the variation in the vertical direction is small relative to the horizontal size scale, and the movement of the water flow in the vertical direction is much smaller than the movement in the horizontal direction (because of the rotation of the earth). Under these two assumptions, the ocean dynamics model in the horizontal plane is described by the two-dimensional Navier- Stokes equation [35]:  
\[\frac{\partial\omega}{\partial t}+\left(\overrightarrow{V}\
abla\right) \omega=\
u\Delta\omega \tag{1}\]  
where \(\overrightarrow{V}=(V_{x},V_{y})\) is the velocity field, \(\
u\) is the fluid viscosity, \(\omega=\frac{\partial V_{y}}{\partial x}-\frac{\partial V_{x}}{\partial y}\) is the vorticity, \(\
abla\) and \(\Delta\) are the gradient operator and Laplace operator respectively. However, this model is computationally intensive and cannot meet real-time requirements. In practice, Navier- Stokes can be expressed as a superposition of single-point vortices, namely the viscous Lamb vortex. The mathematical expression of the Lamb vortex is shown as:  
\[ \begin{split}& V_{x}(\overrightarrow{r})=-\Gamma\frac{y-y_{0}}{2 \pi(\overrightarrow{r}-\overrightarrow{r}_{0})^{2}}\Bigg{[}1-e^{-\left( \frac{(\overrightarrow{r}-\overrightarrow{r}_{0})^{2}}{\delta^{2}}\right)} \Bigg{]}\\ & V_{y}(\overrightarrow{r})=\Gamma\frac{x-x_{0}}{2\pi( \overrightarrow{r}-\overrightarrow{r}_{0})^{2}}\Bigg{[}1-e^{-\left(\frac{( \overrightarrow{r}-\overrightarrow{r}_{0})^{2}}{\delta^{2}}\right)}\Bigg{]}\\ &\omega(\overrightarrow{r})=\Gamma\frac{1}{n\delta^{2}}e^{ \frac{(\overrightarrow{r}-\overrightarrow{r}_{0})^{2}}{\delta^{2}}}\end{split}  \tag{2}\]  
Figure 2: The global path planning results (**a**) the task execution sequence after task planning, and (**b**) the refined global path after global path planning. where \(\overrightarrow{r}_{0}\) is the center position vector of the vortex; \(\Gamma\) and \(\delta\) are the intensity and radius parameters of the vortex. The physical model represented by the Equation (2) is used to analyze the velocity field of the surrounding water flow environment [35]. The position, radius, and velocity of the Lamb vortex in practice are analyzed using data obtained by the horizontal acoustic doppler current profiler (H-ADCP). The number of Lamb vortices and the center of the vortex are estimated using the following method. Assuming that the velocity measured by H-ADCP is the tangential velocity produced by the nearest vortex, the vertical direction of this velocity is the radial direction of the vortex, using adjacent velocity information can obtain the center of the vortex. Figure 3 shows a schematic diagram of a water flow Lamb vortex. Since the AUV rarely changes its depth when navigating in large-scale marine, in this paper, the authors only considered the two-dimensional path of AUV. The B-spline curve was used to represent the two-dimensional local path of AUV.