# Multitarget Search Algorithm Using Swarm Robots in an Unknown 3D Mountain Environment
## 2 Ground Target Search Modeling in an Unknown Mountain Environment  
To better study swarm robot ground target search in an unknown mountain environment, the corresponding environment hypothesis is made, and the corresponding mathematical model is established. The search task is located in an \(a\times b\times c\) mountain environment, which has a horizontal area of \(a\times b\) and a height of \(c\). Among them, the mountain height difference is less than \(c\), and the mountain slope changes continuously and is always less than \(a\) degrees. The task object includes the robots, target, and mountain. Robots are represented as set \(\mathbf{Rob}=\{R_{i}|i=1,2,\ldots,n_{u};30\leq n_{u}\leq 100\}\), where \(R_{i}\) represents a robot, and the target is represented as set \(\mathbf{T}=\{tar|j=1,2,\ldots,n_{T};n_{T}\geq 1\}\). The mountain is discretized in both the horizontal \(x\)- and \(y\)-axes with \(\Delta l\) as the unit distance, and the discrete points obtained are called obstacles. Obstacles are represented as set \(\mathbf{S}=\{obs_{k}|k=1,2,\ldots,n_{s};n_{s}\geq 1\}\). The time \(t\) and the locations of \(R_{i}\), \(tar_{j}\), and \(obs_{k}\) are represented as \(\mathbf{R}_{i}(\mathbf{t})\), \(\mathbf{tar_{j}}\), and \(\mathbf{obs_{k}}\), respectively, and the speed of \(R_{i}\) is \(\mathbf{V_{i}}(\mathbf{t})\). The Euclidean distance between each individual is expressed as follows: the distance between robots \(dr_{i_{1},j_{2}|t}\)=\(\|\mathbf{R_{i_{1}}}(\mathbf{t})-\mathbf{R_{i_{2}}}(\mathbf{t})\|\), the distance between a robot and the target \(dr_{i,j|t}=\|\mathbf{R}_{i}(\mathbf{t})-\mathbf{t}\mathbf{a}\mathbf{r}_{j}\|\), and the distance between a robot and obstacle \(drs_{i,j|t}=\|\mathbf{R}_{i}(\mathbf{t})-\mathbf{obs_{k}}\|\). Within the task environment, the search task can be described as follows: considering that the target reaches the threshold value \(d_{0}\), if there is a robot with a target distance \(dr_{i,j|t}<d_{0}\), it indicates that the target is found. When all targets have been found, the target search task is complete. The robots involved in the search have certain characteristics. Assuming that each robot is isomorphic and the robot velocity \(\mathbf{V_{i}}(\mathbf{t})\) satisfies \(0\leq\mathbf{V_{i}}(\mathbf{t})\leq Vm\), each robot can reach any position close to the ground in the task environment. Considering maximum communication distance \(d_{com}\), maximum obstacle sensing distance \(d_{obs}\), and maximum target detection distance \(d_{tar}\), each robot has the following functions: when \(dr_{i_{1},j_{2}|t}\leq d_{com}\), it can communicate between robots; when \(dr_{i,j|t}\leq d_{tar}\), it can detect the target signal; and when \(drs_{i,j|t}\leq d_{obs}\), according to the slope sensor sense obstacles and a robot's relative slope, a robot can climb slopes less than or equal to \(\beta\) and can drive on slopes less than or equal to \(\alpha\) without rollover, and \(\beta<\alpha\). The target being searched for is stationary on the mountain surface within the mission mountain environment. When searching for a target, each robot can continuously detect the target signal using a sensor. The target signal and \(dr_{i,j|t}\) meet an environmental interference function and should describe the function of the target as a response function [23]. The function can be set as Equation (1):  
\[I_{i,j}(t)=\left\{ \begin{array}{cl}\frac{mQ}{dr_{i,j|t}^{2}}+\eta&drt_{i,j|t }\leq d_{tar}\\ 0&drt_{i,j|t}>d_{tar}\end{array} \right. \tag{1}\]  
where \(I_{i,j}(t)\) represents the target signal detected by \(R_{i}\) at time \(t\); \(\eta\) is the environmental disturbance satisfying the standard normal distribution; \(drt_{i,j|t}\) is the objective existence, which is unknown to the robots; \(m\) is the attenuation coefficient of the environment (\(0<m<1\)); and \(Q\) is the constant signal power of the target.