# Multi-UAV Cooperative Anti-Submarine Search Based on a Rule-Driven MAC Scheme
## 2 Materials and Methods
### Building an Association Rule-Driven TPM Model  
In the mission area, the targets may exist individually or in formations depending on their tasks [11], and the target formation has different distribution rules under different task scenarios. Accordingly, the association rule base is established as follows:  
\[S=\{S_{1},S_{2},\ldots,S_{i},\ldots,S_{n}\} \tag{2}\]  
where \(S\) denotes the association rule base, and \(S_{i}\) is the distribution rules of the targets in one of the tasks. Based on (2), the association rule function for multiple targets is defined as:  
\[c_{k}(S)=\left[ \begin{array}{cccccc}\varepsilon_{1}f_{k}(S_{1})&\varepsilon_{ 2}f_{k}(S_{2})&\ldots&\varepsilon_{i}f_{k}(S_{i})&\ldots&\varepsilon_{n}f_{k}( S_{n})\end{array} \right]^{\mathrm{T}} \tag{3}\]  
where \(c_{k}(S)\) is the association rule for multiple targets at time \(k\), \(f_{k}(S_{i})\) is the target distribution function under association rule \(S_{i}\) at time \(k\), and \(\varepsilon_{i}\in\{0,1\}\), \((i=1,2,\ldots,n)\) represents the binary weight and satisfies \(\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{i}+\cdots+\varepsilon_{n}=1\). When \(\varepsilon_{i}=1\), corresponding to \(\varepsilon_{1}=\varepsilon_{2}=\),\(\ldots,=\varepsilon_{i-1}=\varepsilon_{i+1}=\varepsilon_{n}=0\), the current association rule function selection is \(f_{k}(S_{i})\). After some targets are found, the position coordinates of other targets are updated according to the association rule function, \(c_{k}(S)\). Due to the uncertainty of the movement of the target, the target position can only be obtained with a certain probability based on the association rule function. Therefore, the task area grid is defined as three forms: (1) high-probability area, \(hp\); (2) low-probability area, \(lp\); and (3) uncertain-probability area, \(mp\). The search process uses association rule \(S_{i}\) to describe changes in high and low TPMs, as shown in the following steps. Step 1. In the initial stage, according to the prior information, distribute the high and low TPMs, as shown in Figure 2a;  
Figure 1: Multi-UAV cooperative search target scene. Step 2. If UAVs find targets, according to association rule \(S_{i}\), the TPM in the sea area is updated in real time. For example, after finding target \(A\), according to the characteristics of submarine-guardship formation [11], the TPM in the surrounding grid of \(A\) increases; the update process is shown in Figure 2b;  
Step 3. If UAVs find the second target \(B\), according to association rule \(S_{i}\), the probability of some grids on the target connection line and the vertical line will be upgraded to the high-probability region, \(hp\); the corresponding adjustment TPM is shown in Figure 2d;  
Step 4. UAVs continue to search according to the rule base until all targets in the area are found, and the search task ends. Then, define a matrix with multiple elements to represent the initial information of the search environment as:  
\[\psi=\left[p_{mn}\quad c_{0}(S)\quad f_{g}\right]_{L_{x}\times L_{y}} \tag{4}\]  
where \(p_{mn}\in[0,1]\) is the prior probability of grid \((m,n)\); \(c_{0}(S)\) represents the association rule initialization function; and \(f_{g}\) is the flag bit, which is used to identify whether there is a target in the grid. The value of \(f_{g}\) can be expressed as:  
\[f_{g}=\left\{ \begin{array}{ccc}1&,&\text{target}\\ 0&,&\text{No}\end{array} \right. \tag{5}\]  
Based on (3)-(5), the update matrix of the TPM driven by association rules is formed as:  
\[ \begin{bmatrix}\begin{bmatrix}S_{i}&P_{s}&f_{g}\end{bmatrix}&\cdots&\begin{bmatrix} S_{i}&P_{s}&f_{g}\end{bmatrix}\\ \vdots&\ddots&\vdots\\ \begin{bmatrix}S_{i}&P_{s}&f_{g}\end{bmatrix}&\cdots&\begin{bmatrix}S_{i}&P_{s} &f_{g}\end{bmatrix}\end{bmatrix}  \tag{6}\]  
where \( \begin{bmatrix}S_{i}&P_{s}&f_{g}\end{bmatrix} \) represents the information contained in a grid, \(S_{i}\) is the association rules for the grid, and \(P_{s}\in\{hp,lp,mp\}\) denotes the probability characteristics of the grid under association rule \(S_{i}\).