1. 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

2. Overview  Searching the environment for a moving evader  Application in air traffic control, military strategy and trajectory tracking.

3. Overview  Number of pursuers needed  Information space  Planning paths of pursuers

4. Number of pursuers  Depends on the environment  Assumptions: Evader’s motion is continuous Evader can move arbitrarily fast Pursuers can see in all directions.

5. Effect of Geometry on Number of Robots Two robots are needed

6. Upper Bounds Simply-connectedwith hole

7. Upper Bounds Simply-connected: O(lg n), n is # edges F1F2 Partition into two regions Each partition has >= 1/3 of the edges A triangle needs 1 pursuer. kk (k+1) th

8. Upper Bounds Hole: O(h + lg n), h is # holes, n is # edges Reduce to simply-connected

9. Lower bound  Simply connected: Ω(lg n)  Θ(lg n)  Graph searching: Parsons’ problem 2 3 4

10. Example

11. Upper bound  Hole: Θ(sqrt(h)+lg n) Sqrt(h)+sqrt(2h/3)+sqrt(4h/9)  O(sqrt(h)) Partition into two regions Each partition has <= 2/3 holes

12. Recontamination  1 pursuer but O(n) recontaminations !

13. 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

14. 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

15. 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 (x,y)

16. 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

17. 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

18. Partitioning into Cells  Shoot rays off edges in both directions if possible and from vertices if no collisions in either direction

20. 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

21. Information Space Graph: example 00 01 10 11 0 1

22.  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

23. Example Clean Contaminated Visible

24. In More Detail

25. Re-contamination

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

27. 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

28. Animated Visibility

29. 2 Robots

30. 3 Robots

31. Robot with Cone of Vision