1. Fall 20051 Path Planning from text

2. Fall 20052 Outline Point Robot Translational Robot Rotational Robot

3. Fall 20053 Visibility Graph (Point Robot) start goal Edges between all pairs of visible vertices Use graph algorithm to find a path from start to goal

4. Fall 20054 Free Space (Point Robot)

5. Fall 20055 Path Planning (Point Robot)

6. Fall 20056 Path Planning (cont)

7. Fall 20057 Robot (translational) polygonal

8. Fall 20058 C-space Obstacle of P

9. Fall 20059 Minkowski Sum Coordinate dependent!

10. Fall 200510 Theorem CP is P  (-R(0,0)) R (0,0) –R (0,0) Proof:  R(x,y) intersect P  (x,y)  P  (-R(0,0))

11. Fall 200511 R (0,0) –R (0,0) ( x,y ) If intersect, (x,y) is in CP q

12. Fall 200512 R (0,0) –R (0,0) ( x,y ) If (x,y) is in CP, R(x,y)&P intersect p r

13. Fall 200513 Computing Minkowski Sum

14. Fall 200514 Example v1,v4 v2 v3 w1,w5 w2 w3 w4 [i,j] = (1,1) Add v 1 +w 1 angle(v1v2) > angle(w1w2)  j  2

15. Fall 200515 v1 v2 v3 w1 w2 w3 w4 [i,j] = (1,2) Add v 1 +w 2 angle(v1v2) < angle(w2w3)  i  2

16. Fall 200516 v1 v2 v3 w1 w2 w3 w4 [i,j] = (2,2) Add v 2 +w 2 angle(v2v3) > angle(w2w3)  j  3

17. Fall 200517 v1 v2 v3 w1 w2 w3 w4 [i,j] = (2,3) Add v 2 +w 3 angle(v2v3) < angle(w3w4)  i  3

18. Fall 200518 v4,v1 v2 v3 w1 w2 w3 w4 [i,j] = (3,3) Add v 3 +w 3 angle(v3v4) > angle(w3w4)  j  4

19. Fall 200519 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (3,4) Add v 3 +w 4 angle(v3v4) < angle(w4w5)  i  4

20. Fall 200520 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (4,4) Add v 4 +w 4

21. Fall 200521 Non-convex polygons

22. Fall 200522 Time Complexity It is O(n+m) if both polygons are convex. It is O(nm) if one of the polygons is convex and one is non-convex. It is O(n 2 m 2 ) if both polygons are non- convex.

23. Fall 200523 Example 2 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (1,1) Add v 1 +w 1 angle(v1v2) < angle(w1w2)  i  2

24. Fall 200524 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (2,1) Add v 2 +w 1 angle(v2v3) > angle(w1w2)  j  2

25. Fall 200525 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (2,2) Add v 2 +w 2 angle(v2v3) > angle(w2w3)  j  3

26. Fall 200526 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (2,3) Add v 2 +w 3 angle(v2v3) < angle(w3w4)  i  3

27. Fall 200527 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (3,3) Add v 3 +w 3 angle(v3v4) > angle(w3w4)  j  4

28. Fall 200528 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (3,4) Add v 3 +w 4 angle(v3v4) < angle(w4w5)  i  4

29. Fall 200529 v4,v1 v2 v3 w5,w1 w2 w3 w4 [i,j] = (4,4) Add v 4 +w 4

30. Fall 200530 Rotational Robot R (x, y, Ф) Ф: rotated anti- clockwise through an angle Ф

31. Fall 200531 Rotatonal Robot Motion Plan Piano mover applet

32. Fall 200532 C-space of Rotational Robot

33. Fall 200533 Path Planning (Rotational Robot) Each slice: R(0,0,  i ): obtain a roadmap Project all roadmap to get “ intersection ” – a pure rotation from  i to  j Use a slight larger robot to ensure pure rotation won ’ t collide with obstacles

34. Fall 200534 Homework P R [use the grid line to compute the result as accurate as possible] 1.Compute CP w.r.t. R 2.Compute CP w.r.t. R ’ 3.R and R ’ are exactly the same robot, differ only in reference point. Are CPs in 1 and 2 the same? 4.Do 1 and 2 obtain the same answer regarding to the intersection query? That is, the configuration shown left is reported as intersection in 1 & 2. R’