Searching an Evader in Unknown Environments ...

Searching an Evader in Unknown Environments Using Graph Theory August 23, 2014

Abstract

Imagine two scenarios: (i) a civilian lost in a complex building, (ii) a thief hiding in a museum. For the two cases the searcher must find the person. Because of the increasing interest on the pursuit-evasion applications, such as searching buildings for intruders, traffic control, military strategy, and surgical operation, a lot of research [11, 12, 13, 24] has been done on this problem. The pursuit-evasion game is finding an unpredictable Target in a workspace with obstacles, it is known as one of the fundamental problems studied by robotic researchers. The problem is divided into two parts: find the number of pursuers needed to guarantee that the evader will be found, and find the optimal path of the pursuer(s) such that the evader cannot escape to an explored part of the environment and will be eventually located. This problem can be seen in two cases: Unknown environments [8, 9, 20] and known environment [17].

In this paper, we concentrate our work on a form of pursuit-evasion problem, in which a pursuer must find a strategy of moves in an unknown environment (building) to guarantee the localization of an unpredictable evader, which can move arbitrarily fast. Our goal is to find a technique such that the main pursuer can decide the appropriate action for each position in such way that the evader cannot move from unexplored to explored part of the building, and the evader will be eventually found, the action could be: go forward, request a guardian to secure the current area, or go back and select another path. Our work started with the technique to detect obstacles and critical points in walls; this has been done using a Laser-based sensor. Moreover, the search technique called Direct Strategy has been exposed; this technique is based on the idea of deciding the appropriate action for each position in function of the current state and the history. Simulations and results have been obtained using the NetLogo framework. keyword: pursuit-evasion, robotics, graph theory, Laser-based obstacle detection.

1

In the first section, we show other techniques to solve the pursuit-evasion problem, the second section explains how is the environment represented in our work, third section is for the perception method where we use a laser based sensor to detect walls and critical points, in section four we present the technique used to explore the environment with some examples, and we finish with a conclusion to discuss the obtained results.

Introduction

in this article, we address a form of the problem known as pursuit-evasion game; the goal of this problem is how to organize one or more searchers through a given environment in such way to guarantee that any evader(s) present in the environment will be found. 1

2

2.1

Previous work

Graph-based point of view

The first rigorous formulation of the pursuit-evasion problem is due to Parsons in 1976 [21], he restricted his work to the study of the environment represented as a discrete graph, the evader is assumed to be able to move arbitrarily fast through the graph. Other works [5] used Graphs as a platform to test theories on Pursuit-Evasion Games where a player must be as near as possible from another player, and the second player need to be as far as possible. Recent works [16] are based on an instance of pursuit-evasion wich is a Cop trying to find a gambler in a finite graph using probabilistic distributions.

2.2

Figure 1: A computed clearing trajectory for a πsearcher [10].

Visibility-based point of view

Many variations of the polygon search problem (also called visibility-based problem) [2, 6, 10, 11, 19, 27] have been proposed and studied in the literature since its first proposal by Suzuki and Yamashita [24].

2.3

Detection of mobile intruders in a simple polygon was first considered in the searchlight scheduling problem [23] in which the rays of stationary searchlights are used to find the intruder. The use of a mobile searcher having various degrees of visibility for detecting mobile intruders was then considered as polygon search problem in [24] where a number of necessary conditions and sufficient conditions for given polygon to be searchable by various searchers are presented.

Another point of view is based on randomized algorithms to solve the pursuit-evasion problem [14, 1]. The random-based pursuit-evasion problem used environments represented by graphs and utilises random-based algorithms to find the optimal strategy to locate the intruder(s), so works go further by trying to capture the located intruder [14]. 2

Randomized point of view

In [26] the implemented probabilistic framework was on a team of Unmanned Aerial Vehicle (UAV) and Unmanned Ground Vehicle (UGV). [12] proposed a ”greedy” policy to control a swarm of autonomous agents in the pursuit of one of the several evaders.

Figure 2: Triangulation of a polygon and its dual tree [14].

2.4

Probabilistic point of view

Other works used probabilistic tools to solve pursuitevasion problem [12, 13, 26]. The classical approach to pursuit-evasion games is to first build a map of the terrain and then play the game in a known environment. For the map building stage, several techniques have been proposed, see e.g. [7] and references therein. Most of them are based on Bayesian estimation and are implemented using Extender Kalman Filter. The main problem with these map building techniques is that they are time consuming and computationally expensive, even in the case of simple two dimensional rectilinear environments. On the other hand, most of the literature in pursuit-evasion games, see e.g. [12, 24, 25], assumes worst case motion for the evaders and an accurate map of environment. In [12] the pursuit-evasion game and map building problems are combined in a single probabilistic framework. The basic scenario considers multiple pursuers trying to capture a single randomly moving evader. In [25] we extended the scenario to consider multiple evaders and proposed a single vision-based algorithm for evaders and proposed a simple visionbased algorithm for evader detection.

Figure 3: Pursuit with the constrained greedy policy [12].

2.5

Game theory point of view

Some works [4, 5, 15] model pursuit-evasion as a game of two or more player, where one or more players (pursuer) try to capture an evader which is one player that tries to prevent this capture. in this approach, some works [15] model the environment as a grid of size L × W and the pursuer/evader as a point in cell which can move to the 4/8 neibhour cells or stay in the same cell. 3

(a) World 0

Figure 4: A rectangular grid of size L W with cells [15].

3

World Conditioning (b) World 1

3.1

Environment Figure 5: Examples of environments used in experiments

The used environment in the simulation work is a grid of patches; each patch is either black (empty space) or red (obstacle). Figure 5 shows examples of worlds (Also known as environments and buildings). The black areas are the reachable zones, and the red parts are the walls.

3.2

Vertices

We assume there are specific points (called vertices) of the environment, that are significant in order to solve the pursuit evasion problem. Those points are defined as follows (according to [20]): 4

Definition 1 A vertex is considered as critical if and Definition 2 A vertex sees another vertex when only if the angle formed by this adjacent edges is su- there is a link between these two vertices which is not perior to π. Intuitively, if we could put a pursuer cut by a wall. with an omni-directional vision on each critical vertex, there would have no place where the intruder could hide.

3.3

Visibility graph

Visibility graph is very important in this work, the vertices are connected by links, each link between two vertices defines that these two vertices see each the other. Definition 3 A vertex sees directly another vertex when this one can see completely the second vertex, generally, this vertex can be deleted.

4

Obstacle perception

Figure 6: Visibility graph Figure 6 shows an example of environment after generating the current visibility links from the current position 6; the gray links are the simple visibility links, the red links are visibility links with direct view (as defined in definition 3); the cyan parts of the environment are the visible areas by the agent. it is observed that the agent can see points (1, 2, 3, 9, 10, and 15).

To perceive walls, a method is proposed to detect vertices. This technique is based on laser to detect the distance from agent to wall, and then generate the virtual perception for the current position of the main pursuer [3, 18]. This method is not used in the experimental work, the perception of vertices is supposed done before. 5

angle of the corresponding point (see Figure 7).

5 5.1

Direct strategy technique Definition

In this case, the master has to decide whether to request other agent, or go forward, for each position. In the case where the searcher detects more than one vertex at sight, it thinks that it is appropriate to request agent to guard this area, and then to go forward and continue clearing the building. (a) Point 1

5.2

Algorithm

The main algorithm of decision used by the searcher is as follows: Algorithm 1 Decision scenario in Direct Strategy. If there is one Vertex in CV then: Insert current vertex in stack Move to this new vertex If there is more than one vertex in CV then: If there is no walker controlling this area (including this vertex) then put guardian (b) Point 2 in this area Insert current vertex in stack Figure 7: Laser-Based wall’s view Move to the first vertex in CV If there is no vertex in CV then: Figure 7 is the perception of walls according to Create tmp, the list of vertices in QL the agent. These images are plotted on agent’s mind that are visible from V using the angle and distance from the agent to the If tmp contains only one vertex then: perceived wall for each point. As we can see, the Insert current vertex in stack view is not very clear, but the critical points in walls Move to this new vertex can be detected. This perception is built using the If tmp is empty then Go back following trigonometric laws:  x = d.sinθ N (x, y) = (1) y = d.cosθ Where: Where d is the distance from the agent to the wall; 1 and θ is the angle from the referential north to the 1 The

referential north for the NetLogo is the angle that the

agent has when it is heading up; this angle is referenced by 0.

6

CV stack Clone-walker QL V Kill-walker Go-back

List of visible vertices through V 5.3 List of visited vertices, it is used to go back in case of finishing a path Create a walker on V to guard the area List of non-visited vertices through time Current vertex Dismiss walker in current position Go back to previous vertex in stack

Scenario

this algorithm contains 3 cases:

1. The first case is when the agents detect that there is only one other vertex visible (seen for the first time) from the current position, so the agent moves directly to this point.

Figure 8: Case of visibility checking 2. The second case is when the agent perceives more than one other vertex, so it has to request a guardian for this area, and then it moves to For example, suppose the master is on vertex 7 (see one of these vertices (the other vertices that are Figure 8). In this situation, the CV contains the not visited yet will be put in the queue list so vertices 8, 2, 0, 3, 5, 4, 1. The master has to request they will be visited later). a new agent (walker). And then go to another vertex (in this case, it will go to vertex 8). Before that, it puts vertex 7 in the Stack. And also, it puts other vertices of CV in QL. And then, Master does the same scenario for vertex 8. this technique is based on Depth-First Search (D.F.S) to visit all vertices in visibility graph, the 3. The third case is when the agent does not per- main idea proposed in this work is how to prevent ceive any other vertex, the agent check if it has that the evader escape from an area to another area vertices in queue list, in that case it will do the that has been visited by the pursuer, this idea is to same as case 2. Else, the agent has nothing to put a guardian in each vertex that is connected to do in this area so it goes back. more than one other vertex. 7

5.4

Results

• World 7: [10 8 15 20 3 19 11 0 16 5 1 6 14 13 14 6 1 5 16 7 18 7 16 0 11 12 9 21 9 12 11 19 3 4 17 4 3 20 15 2]

5.5

Used framework

Figure 10 shows the interface implemented in NetLogo, there two stages in the processing of the environment (showing the environment, generating edges), there is 7 different worlds used in the experiments. At the right hand side, there are different monitors showing various variables.

(a) World 0

Figure 10: Used tool

6

Conclusion

As presented, resolving the pursuit-evasion problem is based on graph theory; this case of work concentrates on unknown environments. This work proposed a technique to detect walls and vertices us(b) World 7 ing laser-based sensor, and a technique to check the building with local instantaneous perception of enviFigure 9: Direct search strategies ronment, this technique is Direct Strategy technique, in this approach, the agent decides the appropriate The visited visibles by the main agent, corresponding action for each position. The proposed technique in this paper is based on to Figure 9, are as follows in function of iterations: the D.F.S. (Depth-First Search) to explore Visibility • World 0: [1 2 5 6 8 6 3 4 3 9 3 6 5 7 5] Graph. The Direct Search strategy was very useful to 8

clear unknown environments in minimal time. Unfortunately, this technique has some critical cases where the guardians could be requested uselessly, for example, suppose the agent is in zone with two critical vertices that are in the same area. This technique could be used in rescue issues where minimising clearing time is more important than the needed number of used agents. Future work could be done to improve perception technique, and to apply this technique on real robots.

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