Optimizing Trajectories for Unmanned Aerial Vehicles (UAVs) Patrolling the Border

At first glance, most aspects of border protection activity look like classical examples of zero-sum games, in which the interests of the two sides are exactly opposite. This is how such situations are planned now: this is how border patrol agents are assigned to different segments of the border, this is how routes of coast guard ships are planned, etc. However, there is a big difference between such situations and the traditional zero-sum games: in the traditional zero-sum games, it is assumed that we know the exact objective function of each participant; in contrast, in border protection planning (e.g., in counter-terrorism planning), the adversary’s objective function is rarely known in precise terms; at best, we have the description of this objective function in terms of words from natural language. In this paper, on an example of an UAV patrolling the border, we show how fuzzy techniques can help in planning border protection strategies under such uncertainty. I. PATROLLING THE BORDER: A PRACTICAL PROBLEM Remote areas of international borders can be (and are) used by the adversaries: to smuggle drugs, to bring in weapons. It is therefore desirable to patrol the border, to minimize such actions. Even with the current increase in the number of border patrol agents, it is not possible to effectively man every single segment of the border. It is therefore necessary to rely on other types of surveillance. Unmanned Aerial Vehicles (UAVs) are an efficient way of patrolling the border: • from every location along the border, they provide an overview of a large area, and • if needed at a different location, they can move reasonably fast to the new location, without being slowed down by clogged roads or rough terrain. However, while the area covered by the UAV is large, it is still limited. Due to resource limitations, we cannot have all the points on the border under a constant UAV surveillance. Thus, within a portion of the border that is covered by a UAV, it is necessary to keep the UAV moving. II. HOW TO DESCRIBE POSSIBLE UAV PATROLLING STRATEGIES For simplicity, let us assume that the UAV can fly reasonably fast along the border, so that for each point, the interval Chris Kiekintveld and Vladik Kreinovich are with the Department of Computer Science, and Octavio Lerma is with the Computational Science Program; all authors are from the University of Texas at El Paso 500 E. University El Paso, TX 79968, USA (contact email [email protected]). This work was supported in part by the National Science Foundation grants HRD-0734825 and DUE-0926721 and by Grant 1 T36 GM07800001 from the National Institutes of Health. The authors are thankful to the anonymous referees for their useful suggestions. between two consequent overflies does not exceed the time 2T needed to successfully cross the border area back-andforth. In the ideal case, this would means that the UAV is capable of detecting all adversaries – and thus, preventing all border violations. In reality, however, a fast flying UAV can miss the adversary. It is therefore desirable to select a trajectory that would minimize the effect of this miss. The faster the UAV goes pass a certain location, the less time it spends in the vicinity of this location, the more probable it is that the UAV will miss the adversary. From this viewpoint, an important characteristic of the trajectory is the velocity v(x) with which the UAV passes through the location x. So, by a patrolling strategy, we will mean a function v(x) that describes how fast the UAV flies at different locations. This strategy must be selected in such a way that a total time for a UAV to go from one end of the area to another one is equal to the given value T . The time during which a UAV passes from the location x to the location x + ∆x is equal to