1. Building Surveillance Graphs for GRAPH-CLEAR Andreas Kolling & Stefano Carpin UC Merced Presented by Pras Velagapudi FRC Nav. Reading Group - Nov. 2, 2009

2. Problem: Pursuit-evasion Spatial (typically 2D) world Assumptions about intruders: –Know where robots are –Can move unboundedly Objective: –Find a sequence of actions that detects all intruders using the least number of robots FRC Nav. Reading Group - Nov. 2, 2009

3. GRAPH-CLEAR Formalized subclass of pursuit evasion Environment is weighted graph Two actions –Blocking Robot sits on edge, intruder can’t pass –Sweeping Robot searches vertex, determines if intruder is present FRC Nav. Reading Group - Nov. 2, 2009

4. GRAPH-CLEAR FRC Nav. Reading Group - Nov. 2, 2009 In order to sweep, all edges must be blocked

5. Surveillance Graph Blocking edges –Edges are blocked when their corresponding areas are completely segmented by sensor coverage Sweeping vertices –Any local clearing strategy for region corresponding to vertex –Simple bounding box sweeping used here FRC Nav. Reading Group - Nov. 2, 2009

6. Moving to the real world… FRC Nav. Reading Group - Nov. 2, 2009

7. Moving to the real world… FRC Nav. Reading Group - Nov. 2, 2009 Step 1: Surveillance Graph GRAPH- CLEAR Solver Step 2: ??? Step 3: Profit!

8. Step 1: Surveillance Graph FRC Nav. Reading Group - Nov. 2, 2009 Start with generalized Voronoi graph

9. Step 1: Surveillance Graph FRC Nav. Reading Group - Nov. 2, 2009 Minima requirements: 1 nearby point is farther from Voronoi edge No nearby points are closer to Voronoi edge Find local clearance minima

10. Step 1: Surveillance Graph FRC Nav. Reading Group - Nov. 2, 2009 clearance minima Find local clearance minima

11. Step 1: Surveillance Graph FRC Nav. Reading Group - Nov. 2, 2009 Partition along minima to create initial SG

12. Step 1: Surveillance Graph FRC Nav. Reading Group - Nov. 2, 2009 Compute edge weights w(e) = 2 e

13. Step 2: Improving the graph Auto-generated graph can be inefficient –e.g. GVG + aliasing = extra vertices So, do some optimization! FRC Nav. Reading Group - Nov. 2, 2009

14. Step 2: Improving the graph Collapse leaf nodes Collapse chains FRC Nav. Reading Group - Nov. 2, 2009

15. Step 2: Improving the graph FRC Nav. Reading Group - Nov. 2, 2009

16. Step 3: GRAPH-CLEAR solving Start with weighted surveillance graph Edges = narrow corridors between regions Vertices = wide and open regions Weights = # of robots to sweep/block area Vertices = {Contaminated, Clear} Edges = {Contaminated, Clear, Blocked} FRC Nav. Reading Group - Nov. 2, 2009

17. Step 3: GRAPH-CLEAR solving Shove robots in doors until you have a tree graph Efficient solution for tree graphs FRC Nav. Reading Group - Nov. 2, 2009 NP-complete, however, efficient for trees:

18. Step 3: GRAPH-CLEAR solving Tree solution overview: 1.Compute bidirectional labels for each edge (Labels represent the cost of clearing the subtree on the other side of the edge, if the source node is already cleared) 2.Find lowest cost label 3.Start clearing tree from that label FRC Nav. Reading Group - Nov. 2, 2009

19. Step 3: GRAPH-CLEAR solving FRC Nav. Reading Group - Nov. 2, 2009

20. Step 3: GRAPH-CLEAR solving Leaf node: Internal node: FRC Nav. Reading Group - Nov. 2, 2009

21. Step 3: GRAPH-CLEAR solving Actually, there is an optimal O(n 2 ) algorithm –Compute cuts on the graph for cleared vertices – Subdivide the problem into that of solving the clearing problem for each subtree FRC Nav. Reading Group - Nov. 2, 2009

22. Results UC Merced FRC Nav. Reading Group - Nov. 2, 2009

23. Results Radish (sdr_site_b) FRC Nav. Reading Group - Nov. 2, 2009

24. Results FRC Nav. Reading Group - Nov. 2, 2009 sensing range# initial verts # final verts # robot (initial) # robots (final) # cycles cost of cycles map coverage w/ non-MST robots

25. Conclusion GRAPH-CLEAR is applicable to real-world problems Can construct efficient surveillance graphs using simple methods Future work –Probabilistic variants –Local optimizations in search patterns FRC Nav. Reading Group - Nov. 2, 2009