# Wildlife Monitoring Using a Multi-UAV System with Optimal Transport Theory
## 3 Method
### Other Exploration Strategy: Lawn Mower Method  
For the performance comparison purpose, we introduce another monitoring strategy--lawn mower exploration scheme, one of the most widely used methods to explore the given domain. A description of the lawn mower exploration method is provided below. In the lawn mower monitoring strategy, a single or multiple agents are tasked with exploring an area of interest uniformly in a zigzag manner. For the wildlife monitoring application, the exploration area can be determined from the tracked animal information. If multiple agents are deployed for exploration, then the exploration area is divided equally between multiple agents for independent but balanced exploration. Each agent generates equally spaced horizontal and vertical line segments to create waypoints and explores the assigned region uniformly as depicted in Figure 5.  
Figure 5(a) provides the conceptual drawing to show how the exploration area is determined and Figure 5(b) illustrates the waypoint generation for the two-agent case. Given that \(G\in\mathbb{N}\) numbers of animals are being tracked and the locations of these animals \(\{z_{T}^{d}\}_{q=1}^{G}\) at time \(T\) are known from the GPS trackers (presented as red triangle symbols in Figure 5(a)), the sets of x-coordinates and y-coordinates for these known animal locations are \(X_{a}=\{z_{x}^{1},z_{x}^{2},\ldots,z_{x}^{G}\}\) and \(Y_{a}=\{z_{y}^{1},z_{y}^{2},\ldots,z_{y}^{G}\}\), respectively. Then, the parameters \(X_{min},Y_{min},X_{max},Y_{max}\) to determine the initial search area (rectangular area ABCD in Figure 5(a)) can be calculated by  
\[X_{min} =min(X_{a}), Y_{min} =min(Y_{a}) \tag{11}\] \[X_{max} =max(X_{a}), Y_{max} =max(Y_{a}),\]  
where \(X_{min},X_{max}\in\mathbb{R}\) (or \(Y_{min},Y_{max}\in\mathbb{R}\)) are the minimum and maximum x-coordinates (or y-coordinates) of the initial search area, respectively. In practical scenarios, the base station for the team of UAVs may be located far away from the location of the detected animal herds by the GPS trackers. When the UAVs arrive at the last updated GPS locations, the animals may not be there anymore as they may have moved to another location. Thus, the monitoring domain should be expanded considering the time delay after dispatching a team of UAVs. The expansion will be given in both horizontal and vertical directions in an unbiased manner since the animal movement directions are completely unknown. The parameters \(X_{min}^{\prime},Y_{min}^{\prime},X_{max}^{\prime},Y_{max}^{\prime}\) for the expanded search area (rectangular area A'B'C'D' in Figure 5(a)) can be calculated from  
\[X_{min}^{\prime} =X_{min}-f_{X}(X_{max}-X_{min}) \tag{12}\] \[Y_{min}^{\prime} =Y_{min}-f_{Y}(Y_{max}-Y_{min})\] \[X_{max}^{\prime} =X_{max}+f_{X}(X_{max}-X_{min})\] \[Y_{max}^{\prime} =Y_{max}+f_{Y}(Y_{max}-Y_{min}),\]  
where \(X_{min}^{\prime},X_{max}^{\prime}\in\mathbb{R}\) (or \(Y_{min}^{\prime},Y_{max}^{\prime}\in\mathbb{R}\)) denote the minimum and maximum value of the x-coordinates (or y-coordinates) of the expanded search area, respectively. Moreover, \(f_{X},f_{Y}\in\mathbb{R}\) are defined as the expansion factors in the horizontal and vertical directions, respectively. Figure 5: Schematic for the Lawn Mower exploration strategy with two agents. Once the expanded search area is determined, the area is divided equally based on the number of agents \(n_{a}\in\mathbb{N}\) as shown in Figure 5b. The initial waypoint for each agent can be determined recursively as follows:  
\[x_{0}^{k}=\left[x_{0}^{k-1}+\frac{X_{max}^{\prime}-X_{min}^{\prime}}{n_{a}},Y_{ min}^{\prime}\right]\qquad k=1,2,\ldots,n_{a}, \tag{13}\]  
where \(x_{0}^{k-1}\in\mathbb{R}\) is the x-coordinate of the initial position for \((k-1)\)th agent. For the first agent, \(x_{0}^{k}=X_{min}^{\prime}\).