# Wildlife Monitoring Using a Multi-UAV System with Optimal Transport Theory
## 2 Problem Description and Theoretical Background  
A set of real and natural numbers are denoted by \(\mathbb{R}\) and \(\mathbb{N}\), respectively. Further, \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). The symbols \(\|\cdot\|\) and \({}^{T}\), respectively, denote the Euclidean norm and the transpose operator. The symbol \(\mathcal{R}(x,r)\) represents a set of points within the circle centered at \(x\) with a radius \(r\). The symbol \(\#\) indicates the cardinality of a given set. The variable \(t\in\mathbb{N}_{0}\) is used to denote a discrete-time index. Suppose that a team of UAVs is deployed to monitor wildlife as shown in Figure 1. Due to the time-varying nature of wildlife locations, it is not an easy task to detect animal herds using a team of UAVs even if locational information is available from the GPS trackers. The GPS trackers only provide limited information with intermittent data to save the battery. The size of the domain is another factor obstructing the detection of animal herds as it is in general very spacious. Moreover, UAVs have limited energy and thus, are not able to cover the entire domain because of its huge size. Throughout the paper, we assume that the locations of UAVs are accurately known by GPS signals. Also, it is assumed that the UAVs can detect animals during the monitoring mission via onboard image processing such as machine learning technology. Although the animal recognition and detection problem itself is another important research area for wildlife monitoring, it is out of scope in this study. Rather, we are more interested in which areas the UAVs should cover to increase the wildlife detection rate, which is a challenging problem as stated above. There needs a wildlife monitoring strategy using a multi-UAV system to maximize the wildlife detection rate. In this study, we propose that a team of UAVs search for animal herds reflecting the density distribution that describes the probability of finding animals in the domain. This density distribution can be constructed from the last-received GPS tracker information or satellite images. In this case, the UAVs should spend more time on the high probability area while exploring the low probability area with less time since the probability of the given distribution indicates how likely the UAVs can find animals. As animals do not necessarily stay at one location and move around the domain, the density distribution also needs to change for the spatio-temporal evolution of the distribution. Looking from the above perspective, the proposed wildlife monitoring strategy must address the following research questions: (1) what is the proper metric to measure the similarity between the distribution formed by the trajectories of UAVs and the given density distribution? (2) what is the control method for the team of UAVs to achieve the similarity between the two distributions? (3) how to incorporate the spatio-temporal evolution of the given density distribution for the wildlife movement into the control method? Regarding the first question, we introduce the Optimal Transport (OT) problem. Traditionally, the objective of the optimal transport is to obtain an optimal solution for a resource allocation problem [28], where the focus is to determine how a distribution can be transformed into another distribution with minimum effort. This minimum effort can be quantified using the Wasserstein distance for the continuous marginal case. This metric has been utilized in wide range of dynamical systems including system analysis [29; 30; 31] and controller synthesis [32; 33] problems. The Wasserstein distance [28] of order \(p\) can be written in the following form. Figure 1: Illustration of wildlife monitoring using a team of UAVs.