Visibility-Based Pursuit-Evasion in a Polygonal Environment

L.J. Guibas, J.C. Latombe, S.M. LaValle, D. Lin, and R. Motwani

Finding an Unpredictable Target in a Workspace with Obstacles

S.M. LaValle, D. Lin, L.J. Guibas, J.C. Latombe, and R. Motwani

Presented by Ben Wong, Gregory Kron

Overview

Searching the environment for a moving evader

Application in air traffic control, military strategy and trajectory tracking.

Overview

Number of pursuers needed Information space Planning paths of pursuers

Number of pursuers

Depends on the environment Assumptions:

Evader’s motion is continuous Evader can move arbitrarily fast Pursuers can see in all directions.

Effect of Geometry on Number of Robots

Two robotsare needed

Upper Bounds

Simply-connected with hole

Upper Bounds

Simply-connected: O(lg n), n is # edgesF1F2

Partition into two regions

Each partition has>= 1/3 of the edges

A triangle needs 1 pursuer.

k k

(k+1)th

Upper Bounds

Hole: O(h + lg n), h is # holes, n is # edges

Reduce to simply-connected

Lower bound

Simply connected: Ω(lg n) Θ(lg n) Graph searching: Parsons’ problem

2

3

4

Example

Upper bound

Hole: Θ(sqrt(h)+lg n)

Sqrt(h)+sqrt(2h/3)+sqrt(4h/9) O(sqrt(h))

Partition into two regionsEach partition has<= 2/3 holes

Recontamination

1 pursuer but O(n) recontaminations !

Outline

In fact, finding the minimum number of pursuers is NP-hard

Complete Algorithm for Single pursuer Information space (recontamination) Space partitioning into conservative cells Information space graph

Greedy Algorithm for Multiple Pursuers

Information space

The information space is the set of all the information states of the pursuer(s)

An information state is characterized by: The position of the pursuer(s) The regions where the evader may be (contaminated)

Note: The positions of the evader can be grouped into equivalence classes

Single Pursuer: Information State

We label in binary the gap edges: 0: safe 1:contaminated here:(0,0), (0,1), (1,0) or (1,1)

By knowing the location in the Free Space and the state of the gap edges, we uniquely define the Information State

1 or 0

1 or 0

(x,y)

Changes of Information State

Information state only changes when a gap edge appears or disappears

Conservative Cell Partitioning Keep track of just these transitions to simplify without

losing completeness

Information State: (x1,y1,0,1)Information State: (x2,y2,0,1)Information State: (x3,y3,0,1)Information State: (x4,y4,0)Information State: (x3,y3,0,0)Information State: (x,y,x, x)

Clean

Contaminated

Partitioning into Cells

We partition the free space into convex cells that would correspond to the equivalence classes

The edges of such a partition correspond to visibility changes

Partitioning into Cells

Shoot rays off edges in both directions if possible and from vertices if no collisions in either direction

Information Space Graph

Create/connects all Information States All edge gap contaminated/clean combinations for each point A point with 2 edge gaps will have four nodes (00, 01, 10, 11) in this graph Can grow exponentially (problem of checking opt not even know to be NP)

Keep track of gap edges splitting or merging Connections between Information Space States Number of gaps may change; need to preserve the connectivity Preserve contamination

Information Space Graph: example

00

01

10

11

0

1

Search the graph for a solution (Dijkstra’s Algorithm) Initial State has all contaminated edges (11…) Goal State has all clean edges (00…) Each vertex is only visited once Cost function based on Euclidean distance between points

Information Space Graph: research

Example

Clean

Contaminated

Visible

In More Detail

Re-contamination

Multiple Pursuers

Do one as best you can (greedy algorithm) Add another to cover the missed spaces Less complete, but works pretty well

Conclusion

Works well on the case presented Requires a simple, 2D geometry

A recent work by LaValle et al. allows to have curved obstacles

Information State Graph can be very big A recent work by Sang-Min Park developed a quadratic-cost algorithm

for 1 pursuer

Real-world vision is not perfect Can deal with cone vision

Animated Visibility

2 Robots

3 Robots

Robot with Cone of Vision