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Fall 20051 Path Planning from text

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Fall 20052 Outline Point Robot Translational Robot Rotational Robot

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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

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Fall 20054 Free Space (Point Robot)

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Fall 20055 Path Planning (Point Robot)

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Fall 20056 Path Planning (cont)

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Fall 20057 Robot (translational) polygonal

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Fall 20058 C-space Obstacle of P

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Fall 20059 Minkowski Sum Coordinate dependent!

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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))

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

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

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Fall 200513 Computing Minkowski Sum

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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

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

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

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

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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

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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

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

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Fall 200521 Non-convex polygons

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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.

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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

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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

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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

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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

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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

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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

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

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

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Fall 200531 Rotatonal Robot Motion Plan Piano mover applet

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Fall 200532 C-space of Rotational Robot

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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

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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’

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