1. Pursuit / Evasion in Polygonal and General Regions The Work: by LaValle et al. The Presentation: by Geoff Hollinger and Thanasis Ke(c)hagias

3. The Problem: Pursuit / Evasion in a Polygonal Region The Assumptions: Region is simply connected polygon (no holes) The pursuer has a map There is one pursuer, with 360  vision The evader is captured as soon as seen by the pursuer The evader is arbitrarily fast The evader always knows the pursuer’s position The Desired Solution: An algorithm which gives a motion plan which guarantees capture (if such a plan exists) a “can’t do” output (if a guaranteed capture plan doesn’t exist).

4. Variants Non-polygonal region (e.g. with curved boundary) Map of the region is not available Probabilistic search (capture is not guaranteed but has high probability) Pursuit / evasion on a graph (not a region)

5. Key Concepts The polygonal region is denoted by P. For every point x in P, the visibility polygon is and the invisibility set P–V(x) is the union of several disjoint simple connected polygons. Some of these polygons are clean (i.e. they certainly do not contain the evader) and some are dirty (i.e. they may contain the evader). The boundary of V(x) consists of edges; some of these are edges of the original P ; the remaining are gap edges (facing “free space”)

6. Visibility polygon Invisibility set Gap edges (black is clean, Red is dirty)

8. Given some point x, it will have a V(x), with n associated gaps ( n  0) each of which can be clean or dirty (i.e. the invisible component behind that gap will be clean or dirty). This information can be encoded in an n-long string (say of 0’s and 1’s) which we denote by B(x). Note: B(x) can also be the empty string.

9. Information Space We need an appropriate state space for the problem. We could use (x,S) where x is the position of the pursuer S is the set of dirty points But, we would prefer a discrete state space.

10. Note: when we know x, we also know V(x) and so P–V(x), i.e. the invisible components. And S  P–V(x). So we don’t really need to put S in the state, B(x) suffices (and it is discrete). Also: we can discretize P (break it into cells) provided we do not lose any critical information. Critical information is how gaps change. We need a discretization that preserves this information.

11. Critical Gap Events 1.A gap disappears 2.A gap appears (it gets a 0 label) 3.A gap splits into two gaps (they inherit the parents label) 4.Two gaps merge into a new one (it gets a 1 label if any of the original gaps had a 1) Note: gaps can also change in noncritical ways (continuous transformation) Assumption: we never have an event which involves three gaps simultaneously

12. A gap disappears / appears A gap splits into two / two gaps merge.

13. Conservative Discretization Form a discretization D={D 1,…, D N } by: extending all edges of P (inside P ), extending outward segments from all pairs of vertices (inside P ) and taking all resulting sub-polygons as cells D i of the discretization. This is a conservative discretization, i.e. no critical gap events occur while the pursuer moves inside one of the cells.

14. The rulez: Example:

15. Finally instead of (x, B(x)) use as state (D i, B(D i )) (which takes values in a discrete state space, the information space).

16. Now that we have the state space, we need the state transition function. It will be a state transition graph. We actually have two graphs: G c is the connectivity graph; it has N nodes (one per cell) and its edges follow the connectivity of the cells; it is an undirected graph. G I is the information graph (the state transition graph) nodes: for the i-th cell D i it has 2 ni nodes, where n i is the number of gaps associated with any x in D i edges: they respect critical gap events and information changes. Note: G I is a directed graph.

17. Example 1: Discretized polygon Undirected adjacency graph Directed information graph Example clearing sequence: 1-2 1/1 -> 2

18. Now we can formulate and solve the Pursuit/Evasion problem: In G I, find a (shortest) path which starts from a given “all- dirty” node and ends at some “all-clean” node (provided such a path exists).

19. Example 1: Discretized polygon Undirected adjacency graph Directed information graph Example clearing sequence: 1-2 1/1 -> 2

20. Example 2: Discretized polygon Undirected adjacency graph Directed information graph Example clearing sequences: 1)5-4-3-2 5/1 -> 4/1 -> 3/10 -> 2/0 2)3-4-3-2 3/11 -> 4/1 -> 3/10 -> 2/0

21. Example 3: Discretized polygon Undirected adjacency graph Directed information graph Example clearing sequences: 1)1-2-3-4-5 1/1 -> 2/11 -> 3/1 -> 4/01 -> 5/0 2)4-5-4-3 4/11 -> 5/1 -> 4/10 -> 3/0

22. Example 4: Discretized polygon Example clearing sequences: 1) 1-2-3-4-5 1/1 -> 2/11 -> 3/1 -> 4/01 -> 5/0 2) 7-6-5-4-3 7/11 -> 6/111 -> 5/1 -> 4/01 -> 3/0

23. Example 5: Example clearing sequence: 10-9-8-7-6-5-4-3-2-3-4-5-12-13-18-19-20-19-18-13-14-15-16 NodeInfo State 101 901 8011 7 60111 5 4 3011 201 3001 40011 5 120011 130011 180011 19011 2001 19001 180001 130001 140001 15001 1600 Discretized polygon

24. Example 6: Any path leads to recontamination, for instance: 10-9-8-7-6-5-4-3-2-1-22 NodeInfo State 101 901 8011 7 60111 5 4 3011 2 111 Recontaminated!! Oh no!

25. Example 7: Discretized polygon Undirected adjacency graph Directed information graph Not a chance…can’t clear anything

26. Every node of G I can transit to two other nodes. If we assign equal probabilities to trans’s we get a Markov chain. Its states can be divided into two classes: Transient Persistent Trapping (subset of persistent) Furthermore, some states can be collapsed. Some Markov Chain Connections

27. It might be interesting to address questions such as: Decompose the chain to ergodic classes (connected components) Determine how many trapping classes exist. Is a particular trapping class (the all-clean one) accessible from a particular node? If the pursuer performs a random walk on the graph what is the probability of hitting the trapping class? what is the expected time to hit the trapping class? is there an equilibrium probability distribution? what is the rate of convergence to the equilibrium?

28. Variant 1: Non-polygonal region

30. Variant 2: Map of the region is not available The region A sequence of gap navigation trees: tree2tree transitions take place at critical gap events. A gap can be chased until it disappears; when it reappears it is cleared!!!

31. Questions, Issues etc. Is there an information quantity ? If yes, how does it evolve during the pursuit? Does recontamination help? Can we reduce polygon problem to graph problem? If not exactly, then approximately? Conjecture: if the polygon can be cleared starting from a particular all-dirty state, then it can be cleared starting from any all-dirty state. How to use all this for Ember?

32. Biblio S. M. LaValle, D. Lin, L. J. Guibas, J.-C. Latombe, and R. Motwani. Finding an unpredictable target in a workspace with obstacles. In Proc. IEEE Int'l Conf. on Robotics and Automation, pages 737--742, 1997.Finding an unpredictable target in a workspace with obstacles L. J. Guibas, J.-C. Latombe, S. M. LaValle, D. Lin, and R. Motwani. Visibility-based pursuit-evasion in a polygonal environment. In F. Dehne, A. Rau-Chaplin, J.-R. Sack, and R. Tamassia, editors, WADS '97 Algorithms and Data Structures (Lecture Notes in Computer Science, 1272), pages 17--30. Springer-Verlag, Berlin, 1997.Visibility-based pursuit-evasion in a polygonal environment L. J. Guibas, J.-C. Latombe, S. M. LaValle, D. Lin, and R. Motwani. Visibility-based pursuit-evasion in a polygonal environment. International Journal of Computational Geometry and Applications, 9(5):471--494, 1999.Visibility-based pursuit-evasion in a polygonal environment L. Guilamo, B. Tovar, and S. M. LaValle. Pursuit-evasion in an unknown environment using gap navigation graphs. In Proc. IEEE International Conference on Robotics and Automation, 2004. Under review.Pursuit-evasion in an unknown environment using gap navigation graphs B. Tovar, S. M. LaValle, and R. Murrieta. Locally-optimal navigation in multiply- connected environments without geometric maps. In IEEE/RSJ Int'l Conf. on Intelligent Robots and Systems, 2003.Locally-optimal navigation in multiply- connected environments without geometric maps Great Downloadable Book: Planning Algorithms (by Steven M. LaValle) at http://planning.cs.uiuc. edu/book.pdf http://planning.cs.uiuc Lavalle’s home page: http://msl.cs.uiuc.edu/~lavalle/ Great Downloadable Book