1. Visibility Graph Team 10 NakWon Lee, Dongwoo Kim

2. Robot Motion Planning

4. Configuration Space

5. Shortest Paths

7. Shortest path in road map is not a real shortest path because some arcs are btw nodes that are far apart, whereas others are btw nodes that are close to each other. It is just minimum number of hop. To improve this problem, give each arc a weight corresponding to the Euclidean length, and use graph search algorithm that find the shortest path in a weighted graph. Dijkstra's algorithm But, it is still not a shortest path.

8. Shortest Paths Think of this path as an elastic rubber band. Fix the endpoints at the start and goal position. Try to tighten the rubber band. It will be stopped by the obstacles. The new path will follow parts of the obstacle boundaries and straight line segments through open space.

9. Shortest Paths

12. short cut

13. Shortest Paths

17. Visibility Graph Definition: The visibility graph of s and t and the obstacle set is a graph whose vertices are s and t the obstacle vertices, and vertices v and w are joined by an edge if v and w are either mutually visible or if (v, w) is an edge of some obstacle.

18. Visibility Graph

20. Visible Vertices

22. Visible Definition Two points p and q are mutually visible if the open line segment joining them doesn't intersect the interior of any obstacle.

24. Search Tree

25. Computing the Visibility Graph

29. The question is what are the significant event points, and what happens with each event?

30. Computing the Visibility Graph

31. event

32. Computing the Visibility Graph

33. Dual Arrangement Dual of point D Dual of point C x-coordinate is the slope of dual segment

34. Find Events

35. Angular order By using topological plane sweep, event do not need to be sorted.

36. Topological Plane Sweep

37. Upper horizon tree Lower horizon tree

38. Topological Plane Sweep

42. What Happens at Each Event

43. Possible scenarios

44. What Happens at Each Event

48. Thank You Q&A