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Path Planning from text

Path Planning from text. Outline. Point Robot Translational Robot Rotational Robot. Visibility Graph (Point Robot). goal. Edges between all pairs of visible vertices. start. Use graph algorithm to find a path from start to goal. Free Space (Point Robot). Path Planning (Point Robot).

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Path Planning from text

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  1. Path Planning from text

  2. Outline • Point Robot • Translational Robot • Rotational Robot

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

  4. Free Space (Point Robot)

  5. Path Planning (Point Robot)

  6. Path Planning (cont)

  7. Robot (translational) • polygonal

  8. C-space Obstacle of P

  9. Minkowski Sum Coordinate dependent!

  10. 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. If intersect, (x,y) is in CP q (x,y) R(0,0) –R(0,0)

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

  13. Computing Minkowski Sum

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

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

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

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

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

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

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

  21. Non-convex polygons

  22. 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(n2m2) if both polygons are non-convex.

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

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

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

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

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

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

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

  30. Rotational Robot • R (x, y, Ф) • Ф: rotated anti-clockwise through an angleФ

  31. Rotatonal Robot Motion Plan Piano mover applet

  32. C-space of Rotational Robot

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

  34. Homework • [use the grid line to compute the result as accurate as possible] • Compute CP w.r.t. R • Compute CP w.r.t. R’ • R and R’ are exactly the same robot, differ only in reference point. Are CPs in 1 and 2 the same? • 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. P R R’

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