Visibility-Based Target-Tracking Game: Bounds and Tracking Strategies

Abstract

The art gallery problem or museum problem is a well-known visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. This letter deals with a variant of the art gallery problem in which guards are deployed to track a mobile intruder. We consider a team of free guards equipped with omnidirectional cameras deployed to track a bounded speed intruder inside a simply connected polygonal environment. We present a partitioning technique for any simply connected polygon using its diagonals such that each partition is at most a nonagon. Next, we propose an event-triggered strategy for a single guard to track an intruder around a reflex vertex. Based on the proposed strategy, we formulate the problem of finding the minimum speed of the guard and its corresponding trajectory inside any partition as a convex-optimization problem. The solution to the convex-optimization problem provides an upper bound on the speed of the mobile guard required for persistent tracking of the mobile intruder. Furthermore, we show that the number of guards deployed by the proposed technique belongs to $\mathcal {O}(\frac{2n}{7})$ and $\Omega (\frac{n}{6})$ , which is strictly less than $\lfloor {\frac{n}{3}} \rfloor$ (sufficient and sometimes necessary number of guards required for coverage). Finally, we extend the analysis to orthogonal polygons, and show that the upper bound on the number of guards deployed for tracking is strictly less than $\lfloor {\frac{n}{4}} \rfloor$ which is sufficient and sometimes necessary for the coverage problem.