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Fall 20051 Path Planning from text. Fall 20052 Outline Point Robot Translational Robot Rotational Robot.

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Presentation on theme: "Fall 20051 Path Planning from text. Fall 20052 Outline Point Robot Translational Robot Rotational Robot."— Presentation transcript:

1 Fall Path Planning from text

2 Fall Outline Point Robot Translational Robot Rotational Robot

3 Fall 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 Free Space (Point Robot)

5 Fall Path Planning (Point Robot)

6 Fall Path Planning (cont)

7 Fall Robot (translational) polygonal

8 Fall C-space Obstacle of P

9 Fall Minkowski Sum Coordinate dependent!

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

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

13 Fall Computing Minkowski Sum

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

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

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

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

21 Fall Non-convex polygons

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

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

31 Fall Rotatonal Robot Motion Plan Piano mover applet

32 Fall C-space of Rotational Robot

33 Fall 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 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’


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