## 2 The Improved Potential Field Approach  
The potential field approach is a commonly used method for AUV path planning in a surface-water environment. However, the potential field approach is not cooperative when it is applied to target hunting. In this section, applying the cooperation method into potential field is a novel idea for multi-AUV cooperation target hunting. The flow of the target hunting task is shown in Figure 1. In this approach, when one AUV detects the targets, the multi-AUV system will calculate the distance between the each AUV and the target and request the proper AUVs to accomplish the hunting tasks. At the same  
undetected targets number. The variable \(A\) denotes the area of the environment, while \(A_{\alpha}\) represents the already explored area. Because there are obstacles in the environment, the AUVs need to plan the collision-free paths to complete the task. In this case, the repulsive potential \(U_{\gamma}\) is given by [22]  
\[U_{\gamma}= \begin{cases}\omega_{\gamma D}\frac{1}{H_{D}}+\omega_{\gamma H}\frac {1}{H_{H}}+\omega_{\gamma DD}\frac{1}{H_{DD}},&if\ R_{i}\ continues\ exploring,\\ 0,&if\ R_{i}\ moves\ to\ targets,\\ \frac{1}{2}\eta\left(\frac{1}{d}-\frac{1}{d_{1}}\right)\left|X_{Ri}-X_{Tl} \right|^{2},&if\ R_{i}\ detects\ obstacles.\end{cases}  \tag{3}\]  
where \(\omega_{\gamma D}\), \(\omega_{\gamma H}\) and \(\omega_{\gamma DD}\) are the density weights; and \(H_{D}\), \(H_{H}\) and \(H_{DD}\) are the dispersion degree, homodromous degree and district-difference degree respectively. \(d\) is the nearest distance between the AUV and the detected obstacles. \(d_{1}\) is the influence scope of obstacles. \(\eta\) is a position gain coefficient. The relative distance \(\left|X_{Ri}-X_{Tl}\right|\) between the AUV and target is added to the function, which ensures that the global minimum is only at the target in the entire potential field. Dispersion degree \(H_{D}\) evaluates how close the robots are to each other by distance. If there are \(m\) AUVs in a M \(\times\) N area, the parameter \(H_{D}\) is calculated by a Gaussian function as [23]  
\[H_{D}=e^{-\frac{(\delta-p)^{2}}{2\sigma^{2}}} \tag{4}\]  
where \(\delta\), \(\mu\) and \(\sigma\) are calculated by [24]  
\[\delta=\frac{\overline{D}}{\sqrt{M^{2}+N^{2}}} \tag{5}\]  
\[\mu=\frac{1}{t}\sum_{k=1}^{t}\delta_{k} \tag{6}\]  
\[\sigma=\frac{1}{2}[\max(\delta_{k})-\min(\delta_{k})] \tag{7}\]  
\[\overline{D}=\frac{\sum_{j=1}^{m}\sum_{f=j+1}^{m}D(j,f)}{C_{m}^{2}}=\frac{2}{ m(m-1)}\sum_{j=1}^{m}\sum_{f=j+1}^{m}D(j,f) \tag{8}\]  
where \(D\) is the real-time average distance between the AUVs; and \(D(i,f)\) is the distance between AUV \(R_{j}\) and \(R_{f}\). Homodromous degree \(H_{H}\) evaluates how close the AUVs are to others by direction. If there are \(m\) AUVs and the AUV directions are \(\{\theta_{1},\ \theta_{1},\ldots,\theta_{m0}\}\), where \(0^{\circ}\leq\theta\leq 360^{\circ}\). The parameter \(H_{H}\) is calculated by [25]  
\[H_{H}=\frac{\frac{2}{m(m-1)}\sum_{j=1}^{m}\sum_{f=j+1}^{m}abs\left(\theta_{j},\theta_{f}\right)}{m_{0}} \tag{9}\]  
where \(m_{0}\) is the number of possible moving directions of the AUV. In this study, each possible direction area is regarded as a bound area of \(45^{\circ}\) angle. Therefore, there are eight possible direction areas in the simulations. The function abs() is absolute value function. The district-difference degree is used to judge whether all the AUVs stay in the same area. Especially when both \(N_{u}\) and \(A_{u}\) are large, the district-difference degree is applied to provide a proper repulsive potential value to keep the AUVs from gathering. In the actual search task, the environment is usually divided into different parts based on the number of targets and search resources. If both the percentages of undetected targets (denoted as \(N_{u}/N\) ) and unexplored area (denoted as \(A_{u}/A\)) are high, the district-difference degree can help the AUVs to explore separately rather than gather too close to each other.