A Boolean Control Network approach to pursuit evasion problems in polygonal environments

J. Thunberg and X. Hu is with the Division of Optimization and Systems Theory, Department of Mathemathics, Royal Institute of Technology (KTH), Stockholm, Sweden SE-100 44, johan.thunberg@math.kth.se ¨ P. Ogren is with the Department of Aeronautical and Systems Technology, Swedish Defence Research Agency (FOI), Stockholm, Sweden, SE-164 94,
¨ Johan Thunberg, Petter Ogren and Xiaoming Hu
Abstract— In this paper, the multi pursuer version of the pursuit evasion problem in polygonal environments is addressed. This problem is NP-hard, and therefore we seek good enough, but not optimal solutions. By modeling the problem as a Boolean Control Network, we can efficiently keep track of which regions are cleared, and which are not, while the input nodes of the network are used to represent the motion of the pursuers. The environment is partitioned into a set of convex regions, where each region correspond to a set of nodes in the network. The method is implemented in ANSI C, and efficiently solves complex environments containing multiple loops and requiring so-called recontamination. The provided examples demonstrate the effectiveness of the method in terms of computational time.
using a randomized search strategy can locate an evader in any simply connected environment with high probability. The conservative approach is used in the so-called GRAPHCLEAR problem, described below. In the GRAPH-CLEAR problem, [8], the evader and pursuers move in a graph, where each vertex corresponds to a room, and each edge corresponds to a door. Each vertex and edge furthermore has a number assigned to it, corresponding to how many pursuers are needed to clear the vertex (room), or block the edge (door). The problem is now to deploy pursuers to the edges and vertices in such a way that the whole graph is cleared. It is easy to see how most polygonal environments can be divided into rooms and doors. The resulting problem is however, as noted above, a conservative version of the polygonal pursuit evasion problem, as the possibility of seeing from one room to another is removed. In order to balance solution quality and computational performance, one could imagine a hierarchical approach with a global GRAPHCLEAR problem and a polygonal pursuit evasion problem for each room. The approach proposed here is also somewhat conservative, and builds on a framework presented in [11]. However, instead of using Receding Horizon Control (RHC) and Mixed Integer Linear Programming (MILP), a Boolean Control Network [1], [2] is used to formulate and solve the problem. A certain state in this Boolean Control Network corresponds to the environment being cleared, and a successful search strategy thus corresponds to finding pursuer paths such that this state is reached. The notion of visibility that is used is somewhat more conservative than the ’exact’ one used in [4] but still less conservative than the one used in the graphsearch approaches [8]. The main contribution of the paper is that we model the visibility based pursuit evasion game as a Boolean Control Network problem. To the best of our knowledge, this has not been done before. The outline of the paper is as follows. In Section II the polygonal pursuit evasion problem is formally stated and in Section III the proposed solution is described. Section IV contains simulation examples to illustrate the approach and Section V concludes the paper. II. P ROBLEM FORMULATION Following Guibas et al. [4], the pursuers and evader are modeled as points moving in the polygonal free space, F .
I. I NTRODUCTION In this paper we address the visibility based pursuit evasion problem. The aim is to find a search strategy for a group of pursuers, such that an evader moving arbitrarily fast, and starting in an unknown location, will be captured (or seen) no matter what path he decides to take. This problem was first proposed by Suzuki and Yamashita [10] and later studied in e.g., [3], [4], [6], [7]. There are a number of real life applications where the pursuit evasion problem is relevant. For example, when closing a store, the staff needs to make sure that there are no more customers in the facility, and when a burglar alarm has been set off, security personnel must make sure that there are no intruders hiding in the area. The approach is also relevant in some non-adversarial situations, for example when looking for missing children in a warehouse. The general pursuit evasion problem is NP-hard [4] and thus very computationally complex. However, a complete solution for the one pursuer case has been proposed [4], and later extended to incorporate limited field of view [3]. Extending these approaches to the multi pursuer setting has been investigated, but even the two pursuer case turned out to be computationally challenging, [4]. Two ways of dealing with NP-hard problems are to either use a probabilistic setting, or to create a related, more conservative problem, from which valid solutions to the original problem can be constructed. The probabilistic approach was investigated in [7], where it was shown that a single pursuer