Pursuit / Evasion in Polygonal and General Regions The Work: by LaValle et al.

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 setP–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. Invisibility set Visibility polygon 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 • A gap disappears • A gap appears (it gets a 0 label) • A gap splits into two gaps (they inherit the parents label) • 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={D1,…, DN} by: • extending all edges of P (insideP), • extending outward segments from all pairs of vertices (insideP) • and taking all resulting sub-polygons as cells Di 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 (Di, B(Di)) (which takes values in a discrete state space, theinformation 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: • Gc is the connectivity graph; it has Nnodes (one per cell) and its edges follow the connectivity of the cells; it is an undirected graph. • GI is the information graph (the state transition graph) • nodes: for the i-th cell Di it has 2ninodes, where ni is the number of gaps associated with any x in Di • edges: they respect critical gap events and information changes. • Note: GI is a directed graph.

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

18. Now we can formulate and solve the Pursuit/Evasion problem: • In GI , 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: Undirected adjacency graph Discretized polygon Example clearing sequence: 1-2 1/1 -> 2 Directed information graph

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

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

22. Example 4: 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 Discretized polygon

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 Discretized polygon

24. Example 6: Any path leads to recontamination, for instance: 10-9-8-7-6-5-4-3-2-1-22 Recontaminated!! Oh no!

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

26. Some Markov Chain Connections • Every node of GI 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.

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. • 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. • 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. • 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. • 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. • Great Downloadable Book: Planning Algorithms(by Steven M. LaValle) at http://planning.cs.uiuc. edu/book.pdf • Lavalle’s home page: http://msl.cs.uiuc.edu/~lavalle/ Great Downloadable Book