1. Buron Florian CS223b Target finding 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. pursuer evader

2. Buron Florian CS223b Objectives How many pursuer do we need to be sure to find the target? Path planning algorithm of the pursuer? Applications in air traffic control, military strategy, trajectory tracking, building security systems, etc.

3. Buron Florian CS223b Number of pursuer Depends on the environment. Assumptions: Evader’s motion is continuous. Evader can move arbitrarily fast. Pursuers can see in all directions. pursuer

4. Buron Florian CS223b Effect of the geometry on the number of pursuer We need 2 pursuer! Simply-connected edges With an hole

5. Buron Florian CS223b Number of pursuers Finding the minimum number of pursuers needed is an NP -hard problem. Simple upper bound estimation: N=O(log(n)+h), where: N=number of pursuer n=number of edges h=number of holes

6. Buron Florian CS223b Proof for no holes (h=0) N=O(log(n)) Partition into two regions such that each partition has at least 1/3 of the edges Number of edge in each Partition decrease exponentially, and we can ‘reuse’ pursuer

7. Buron Florian CS223b Proof with holes N=O(log(n)+h) Reduce to simply-connected by adding one fix pursuer per hole

8. Buron Florian CS223b Number of pursuers Good estimation? Yes because: Real number of pursuer NP-hard h and n are known Cases were estimation is the real number of pursuer: 2 3 4 1

9. Buron Florian CS223b A Complete Algorithm for Single pursuer

10. Buron Florian CS223b 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)

11. Buron Florian CS223b Information State Example of an information state = (1,1,0) 0 : the target does not hide beyond the edge 1 : the target may hide beyond the edge

12. Buron Florian CS223b Cells decomposition: Critical Curves Information state only changes when a gap edge appears or disappears. Between critical lines the information state is unchanged

13. Buron Florian CS223b Partitioning into Cells Shoot rays off edges in both directions (if possible) and outside of pairs of vertices (only if it is possible for both)

14. Buron Florian CS223b The algorithm Create the cells. Search the graph of cells for a solution. Initial State has all contaminated edges (11…) Goal State has all clean edges (00…) Jump from one cell to one of the adjacent one (check change of information state by keeping track of gap edges splitting or merging) Cost function based on Euclidean distance between points Transform the sequence of adjacent cells in the final path by connecting the centroids of the cells

15. Buron Florian CS223b Example Visible Clear Contaminated

16. Buron Florian CS223b Linear # of Recontaminations Recontaminated area

17. Buron Florian CS223b Example with Two Robots (Greedy algorithm)

18. Buron Florian CS223b Example with Three Robots (Greedy algorithm)

19. Buron Florian CS223b Extension: Robot with Cone of Vision

20. Buron Florian CS223b Conclusion Complete planner in environments searchable by one robot. Planner is rather fast in practice, but its worst-case running time is exponential in n Greedy algorithm for environments requiring multiple robots. But no guarantee of optimality for number of robots. Extensions: cone of vision, aerial robot…