Motion Planning. O Rourke, Chapter 8

Transcription

1 O Rourke, Chapter 8

2 Outline Translating a polygon Moving a ladder

3 Shortest Path (Point-to-Point) Goal: Given disjoint polygons in the plane, and given positions s and t, find the shortest path from s to t that avoids the polygons. t s

4 Any Path (Shape-to-Point) Goal: Given disjoint polygons in the plane, and given a shape S and a position t, determine if there is any path taking S to t avoiding the polygons. t S

5 Any Path (Shape-to-Point) Goal: Given disjoint polygons in the plane, and given a shape S and a position t, determine if there is any path taking S to t avoiding the polygons. Approach: Dilate the polygons by S t to transform this into a point-to-point problem. If s can reach t while avoiding the dilated polygons, S can reach t. s

6 Any Path (Shape-to-Point) Goal: Given Note disjoint that polygons in this case: in the plane, and given a shape SSand is symmetric. a position t, determine if there is any path taking s is Sthe to center t avoiding of S. the polygons. Approach: The traced out boundary self-intersects. Dilate the polygons by S t to transform this into a point-to-point problem. If s can reach t while avoiding the dilated polygons, S can reach t. s

7 Minkowski Sums Definition: Given two point-sets in the plane, A and B, the Minkowski Sum is the set of points: A B = {a + b a A and b B} Note: The sum depends on the choice of origin. A B B

8 Minkowski Sums Definition: Given a point-set A in the plane, its negation is the set of points: A = { a a A} A A

9 Claim: Given shapes A and B, the translation of A by τ intersects B if and only if τ B ( A). A B

10 Claim: Given shapes A and B, the translation of A by τ intersects B if and only if τ B ( A). A B

11 Claim: Given shapes A and B, the translation of A by τ intersects B if and only if τ B ( A). Proof: The translation of A by τ intersects B. a A, b B such that a + τ = b. a A, b B such that τ = b a. τ B ( A).

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13 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) A A B

14 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b

15 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b a b + 1

16 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b b + 1 a

17 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 2 a b + 1

18 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 1 a b

19 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 2 b + 1 a

20 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b b + 1 a

21 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) a b + 1 b + 2

22 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b a b + 1

23 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 1 a b + 2

24 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b a b + 1

25 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b b + 1 a

26 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 2 a b + 1

27 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) a b + 1 b

28 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 2 b + 1 a

29 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 1 b a

30 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 1 a b + 2

31 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b a b + 1

32 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) b + 1 a b + 2

33 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 )

34 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b ) Note:» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) If A is convex, the for loop cycles around the polygon B once. Otherwise we can cycle as many times as there are different extremal vertices in A -- O( A ).

35 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b ) Note:» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) In the worst case, we can trace the next extremal vertex in O( A ) time. If A and B are convex, then in all these tracings of, we cycle around A once -- O 1 time.

36 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: Trace a along edge b, b + 1 a TraceNextLocalExtremum( b + 1 ) Complexity: If A and B convex: O( A + B ) Else if A convex: O( A B ) Else: O( A 2 B ) [I think.]

37 MinkowskiTrace( A, B ) Place A at a vertex of b B. for each a Extremal( b )» for b B: If A or BTrace is not aconvex, along edge to get b, the b + Minkowski 1 Sum, a TraceNextLocalExtremum( b + 1 ) we need to remove the internal polygons. We Complexity: can do this, e.g., by constructing a trapezoidalization, considering If A and B the convex: dual graph, O( A and + B ) using the windingnumber/parity-test to mark the external cells. Else if A convex: O( A B ) Linking the Else: O( A 2 dual cells that are external, we need to determine B ) [I if think.] s and t are in the same connected component.

38 Outline Translating a polygon Moving a ladder

39 Moving a Ladder Goal: Given a line segment at position s, a set of polygons P, and a target position t, determine if the ladder can be moved (translated and rotated) to t while avoiding the polygonal obstacles. t s

40 Moving a Ladder General Approach: Compute the Minkowski Sum of P with l. t s

41 Moving a Ladder General Approach: Compute the Minkowski Sum of P with the ladder. Do this for every rotation of the ladder. t s

42 Moving a Ladder General Approach: Compute the Minkowski Sum of P with the ladder. Do this for every rotation of the ladder. t s

43 Moving a Ladder General Approach: Compute the Minkowski Sum of P with the ladder. Do this for every rotation of the ladder. t s

44 Moving a Ladder General Approach: Compute the Minkowski Sum of P with the ladder. Do this for every rotation of the ladder. Stack into a 3D volume and test if there is a path from s to t, in the 3D space, t that avoids all the polygons. s

45 Moving a Ladder General Approach: Compute the Minkowski Sum of P with the ladder. There are infinitely many angles. Do this for every rotation of the ladder. The boundaries of the 3D sums are not planar. Stack into a 3D volume and test if there is a path from s to t, in the 3D space, t that avoids all the polygons. s

46 Moving a Ladder Implementation: Partition empty region into trapezoids: Base aligned with the ladder direction. Sides defined by the edges of the polygons. In general, as we rotate the ladder, the shape of the cells change but the topology of the partition (i.e. dual graph) does not. At discrete events, cells may be inserted, deleted, or merged.

47 Moving a Ladder Implementation: Partition empty region into trapezoids: Base aligned with the ladder direction. Sides defined by the edges of the polygons. These events can only occur when the rotated ladder aligns with a segment between two vertices of the polygons. Instead of trying all possible rotations, we only need to consider the O( P In general, as we rotate the ladder, the 2 ) cases. shape of the cells change but the topology of the partition (i.e. dual graph) does not. At discrete events, cells may be inserted, deleted, or merged.

48 Moving a Ladder a d f e b c

49 Moving a Ladder a d f e b c

50 Moving a Ladder (a, b) (a, d) (c, d) (e, b) (f, b) (c, b) (a, d) a (a, b) (e, b) e d f (f, b) b (c, d) c (c, b)

51 Moving a Ladder (a, b) (a, b) (a, d) (c, d) (e, b) (f, b) (a, d) (c, d) (e, b) (f, b) (c, b) (c, b) a (a, b) (e, b) d e b (a, d) (c, d) f (f, b) c (c, b)

52 Moving a Ladder (a, b) (a, b) (a, b) (a, d) (c, d) (e, b) (f, b) (a, d) (c, d) (e, b) (f, b) (a, d) (c, d) (e, b) (c, b) (c, b) (c, b) a (a, b) (e, b) (c, d) (a, d) d e f (c, b) b c

53 Moving a Ladder (a, b) (a, b) (a, b) (a, b) (a, d) (c, d) (e, b) (f, b) (a, d) (c, d) (e, b) (f, b) (a, d) (c, d) (e, b) (c, d) (e, b) (c, f) (c, b) (c, b) (c, b) (c, b) a (a, b) (e, b) d e b (c, d) f (c, b) (c, f) c

54 Moving a Ladder Approach: Set Θ = θ 0 = 0, θ 1,, θ n to be sorted angles, with θ i the angle of the i-th segment. For 0 i n:» Compute the Minkowski Sum of P with the rotation of the ladder by angle θ i + ε.» Compute the dual graph of the Minkowski Sum.» Link the i 1 -st and i-th dual graphs. Test if there is a path between the cells containing s and t (at θ = 0) in the linked graph.