1. Visibility-based Motion Planning Lecture slides for COMP 790-058 presented by Georgi Tsankov

2. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

3. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

4. Art gallery problem Question: how many cameras do we need to guard a polygonal art-gallery ? NP-hard Bounds: O(n/3), examples http://www.cs.wustl.edu/~pless/506/l6.html

5. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

6. Sensor Placement Definition Art gallery + more visibility constraints [1]:  Free path  Range constraint (min, max)  Incidence constraint

7. Sensor Placement Algorithm Also NP-hard, so a good and fast approximation is needed. Random Sampling:  Sample M random points and decompose the boundary in R 1,… R N – intervals visible by different sets of points.  Optimal set cover problem – also NP-hard, but efficient approximations exist.

8. Sensor Placement Set cover problem

9. Sensor Placement Inspection Routes Extension: generating inspection routes. [7]  Cost of camera vs. cost of movement Definition: Compute the shortest path for a single camera to inspect the whole boundary. Optimal solution for simple polygons: O(N 4 ) NP-hard when there are holes!

10. Sensor Placement Inspection Routes Approximate algorithm:  Repeat until the boundary is covered Choose an unguarded point P i on the boundary and build its visibility region V(P i ) Sample K points inside V(P i ) and pick the one with the best gain.  Connect the points: TSP with triangle inequality - a 2-approximate algorithm.  Extension to 3D Very hard to compute visibility polyhedron! Simplifications

11. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

12. Map Generation Definitions Goal: Build a map of unknown environment. [9] On-line version of the Sensor Placement problem Partial map, disconnected. Problem: Given a partial map, where should we go next (NBV) ?

13. Map Generation Next Best View How do we compute the next position?  Set of candidates on the free boundary  value = -motion cost + visibility gain

14. Map Generation Example

15. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

16. Finding a Target (pursuit-evasion) Definitions Problem: Given: [3]  A known 2D polygonal environment with obstacles  A target robot with unrestricted speed and unknown initial position plan the movement of a pursuer, so that it eventually sees the target. Applications:  Surveillance  Search and rescue operation  Search for other autonomous robots

17. Finding a Target (pursuit-evasion) Definitions Contaminated/Clear regions Information state: (q, S), the goal is (q, {})

18. Finding a Target (pursuit-evasion) Definitions Visibility region: V(q), gap edges, solid edges. B(q) = (0,1,0,…) - binary vector describing the information state.

19. Finding a Target (pursuit-evasion) Definitions Discretize the space Conservative regions  All points inside see the same edges. (the information state stays the same) How do we build them?

20. Finding a Target (pursuit-evasion) Conservative regions Constructing the conservative regions Put lines whenever a new obstacle edge enters/leaves the visibility region.  Extend all edges  Extend pairs of vertices

21. Finding a Target (pursuit-evasion) Information graph Once we have the regions, build a graph G, and information graph G I :

22. Finding a Target (pursuit-evasion) Information graph How do we compute the edges ?  A gap edge disappears – ok  A gap edge appears – make it clear (0) q1->q2 – a gap edge disappears q2->q1 – a gap edge appears

23. Finding a Target (pursuit-evasion) Information graph Computing the gap transitions (cont’d):  Gap edges merging - OR them  One gap edge splits – assign the same value q3->q4 – two gap edges merge q4->q3 – a gap edge splits

24. Finding a Target (pursuit-evasion) Traversing the graph Just find a path to some goal node (00..0) This is a complete algorithm for one pursuer! Assumes unrestricted visibility.

25. Finding a Target (pursuit-evasion) Multiple pursuers What if one pursuer can’t make the job? How many pursuer do we need: H(F) ?  Depends on the geometry of F.  Upper bound O(sqrt(H) + log(N)) H(F) can not be computed exactly.

26. Finding a Target (pursuit-evasion) Bounded target velocity Evader with bounded velocity [2]  The contaminated regions are more restricted

27. Finding a Target (pursuit-evasion) Summary For 2D:  Complete algorithm for 1 pursuer (with ideal visibility).  Bounds for number of pursuers, strategy for multiple observers. For 3D:  How to compute visibility polyhedron ?  Are gap transitions the same as in 2D ? Demos - http://robotics.stanford.edu/groups/mobots/pe.html http://robotics.stanford.edu/groups/mobots/pe.html

28. Topics Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

29. Following a Target Applications Applications:  Camera movement in VE  Surveillance  Monitoring (automated) processes  Medicine  Military needs Relation to Game theory (the target actively tries to escape visibility)

30. Following a Target Definitions Visibility constraints: free space, range, incidence Visibility sweeping line – the line passing through the target and a reflex vertex. [1]

31. Following a Target Types of problems Critical vs. Average tracking  Critical: Must never lose the target (can’t find it later). The goal is to maximize the escape time.  Average: OK to lose it (probability to re-acquire it at some time). The goal is to maximize the average visibility in some time interval.

32. Following a Target Types of problems Predictable vs. Unpredictable target motion. [4]

33. Following a Target Known target motion Algorithm for predictable motion:  For each time step t, determine V(t) – set of points seeing the target.  The pursuer should be in V(t) for all t.  Restrict V(t) to points, reachable from V(t-1) in one time step  Do this for all t. Complete algorithm, but exponential on dimensions. The space must be discretized.

34. Following a Target Known target motion

35. Following a Target Unpredictable target Unpredictable target motion  No complete algorithm, need approximation  The time horizon is reduced. (one time step). Average vs. Critical tracking

36. Following a Target Unpredictable target – average tracking Maximizing the probability of visibility in the future (one) time steps.  Random motions for the target are considered (heading, velocity) for the time step  Disc with uniform density  Random next positions for the observer are sampled and intersected with the disc.  Choose the best one – maximizes visibility in the next time step.

37. Following a Target Unpredictable target – critical tracking Maximizing the escape time  Shortest distance to escape (SDE)  Select next position, which maximizes SDE

38. Following a Target Other problems Other kinds of problems:  Unknown environment [5]  Stealth tracking [8]  Multiple observers / targets [6]

39. Following a Target Unknown environment Unknown environment – have only local map Build Escape Path Tree (EPT)

40. Following a Target Unknown environment Assign “Escape Risk” to each free edge. Local coordinate system:  n: towards the occluding vertex  t: orthogonal, increases SDE Compute the gradient which minimizes the risk

41. Following a Target Unknown environment The direction for the observer is the average gradient over all escape paths Benefits of the tree (EPT)

42. Following a Target Stealth tracking Stealth tracking  The observer tries to stay outside of the target’s visibility region.  Lookout region.  Escape risk.

43. Following a Target Multiple observers / targets Multiple observers / targets Most of the algorithms can be reused. In [6] the algorithm is exponential on the number of observers. Choosing a metric: worst vs. average risk.

44. Following a Target Summary What we have:  Complete algorithm for predictable target.  Good approximations for unpredictable target.  Algorithm for multiple observers/targets is not efficient enough, and is centralized. Future work:  3D environments  Improved algorithm for multiple observers.

45. Topics covered Art gallery problem Sensor placement Map generation Finding a target (pursuit-evasion) Following a target

46. Acknowledgments Pictures were taken from:  www.cs.cmu.edu/~motionplanning/lecture/Chap6- CellDecomp_howie.pdf www.cs.cmu.edu/~motionplanning/lecture/Chap6- CellDecomp_howie.pdf  Lectures from Latombe’s course: http://robotics.stanford.edu/~latombe/cs326/2004/schedule. htm http://robotics.stanford.edu/~latombe/cs326/2004/schedule. htm  The papers on the next page.

47. References 1. Motion Planning: Recent Developments. - Gonzalez Banos, D. Hsu, Latombe (pp. 19-32). 2. Visibility-Based Pursuit-Evasion with Bounded Speed. - B. Tovar, S. LaValle. 3. Finding an Unpredictable Target in a Workspace with Obstacles. - S. LaValle, D. Lin, L. Guibas, Latombe, R. Motwani. 4. Motion Strategies for Maintaining Visibility of a Moving Target. - S. LaValle, H. Gonzalez-Banos, C. Becker, Latombe. 5. Real-time Combinatorial Tracking of a Target Moving Unpredictably Among Obstacles. - H. Gonzalez-Banos, D. Hsu, Latombe. 6. A Sampling-Based Motion Planning Approach to Maintain Visibility of Unpredictable Targets. - R. Murietta-Cid, B. Tovar, S. Hutchinson. 7. Randomized Planning for Short Inspection Paths. - T. Danner, L. E. Kavraki. 8. Stealth Tracking of an Unpredictable Target among Obstacles. - T. Bandyopadhyay, Y. Li, M. H. Ang, D. Hsu. 9. Navigation Strategies for Exploring Indoor Environments. – H. Gonzalez- Banos, Latombe 10. Visibility-Based Pursuit-Evasion in Three-Dimensional Environments. – S. Lazebnik