1. May 20121 Motion Planning Shmuel Wimer Bar Ilan Univ., Eng. Faculty Technion, EE Faculty

2. May 20122 Outline Problem definition Point robot Work space and configuration space Minkowski sums Translational motion planning Rotational motion planning

3. May 20123 Types of Robots and Motions Articulated Robot Translation Motion start goal

4. May 20124 Rotational Motion start goal Some robots can move in any direction (e.g., ants) Some robots cannot translate (e.g., cars) We’ll study translational and rotational motions Given a robot, is there a free paths (no collisions) from start to goal?

5. May 20125 Work Space and Configuration Space Reference Point 2 Degrees of freedom 3 Degrees of freedom

6. May 20126 Configuration space Work space

7. May 20127 Free Space Computation Divide into trapezoids. It takes O(nlogn) expected time. Remove trapezoids of obstacle in O(n) time.

8. May 20128 Building a Road Map Allocate node at center of vertical edges Allocate node at center of every trapezoid Connect center nodes to edge nodes Done in O(n) with doubly-connected edge list

9. May 20129 Computing a Path Get from start to center of trapezoid in O(logn) time Get from goal to center of trapezoid in O(logn) time Connect center of trapezoids by BFS in O(n) time

10. May 201210

11. May 201211 Convert a problem with polygonal robot into point robot by modifying the obstacles in the configuration space to incorporate the geometry of the robot. Minkowski Sums obstacle robot

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15. May 201215 Extreme Points and Directions

16. May 201216 How complex is Minkowski sum of two polygons?

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18. May 201218 -R a b c d a b c d a b c d a b c d a b c d a b c d Construction of Minkowski Sum 1 2 3 4 5 P Edges of P and R are labeled counterclockwise b 1 2 3 4 5 c d d c a a b

19. May 201219 1 2 3 4 5 a b c d 1 2 3 4 5 P -R a b c d

20. May 201220 b 1 2 3 4 5 c d d c a a b 1 2 3 4 5 a b c d

21. May 201221

22. May 201222 Rotation – Moving a Ladder Minkowski sum for ladder at 0º rotation. Blockage exists.

23. May 201223 Minkowski sum for ladder at 30º rotation. Blockage exists. Minkowski sum for ladder at 60º rotation. Blockage changed.

24. May 201224 Conversion to 3D Motion Problem Bottom view Front view. θ varies from 0º to 75º. Ladder’s reference point can move in the 3D space!

25. May 201225 Cell Decomposition Minkowski sums A B C Cell decomposition 10 8 9 2 1 3 4 5 6 7 Obstacles ∞ R Ladder

26. May 201226 A Cell is the collection of all free points labeled with the same front/back edge label pairs. –A: (3,2); B: (3,8); C: (1,9) Cell decomposition has discontinuities when ladder is oriented similar to an edge. There are finite number of ladder rotation where cell decomposition is changing. –New cells can appear and old ones may disappear.

27. May 201227 5 6 7 10 8 9 2 1 3 4 ∞ R A B C disappeared Ladder is rotated

28. May 201228 In Connectivity Graph G θ nodes are cells of decomposition and edges are connecting nodes corresponding to adjacent cells in free area (a kind of dual graph). A: (3,2) B: (3,8) (1,8) C: (1,9) (1, ∞ ) (10, ∞ ) (5, ∞ ) (3, ∞ ) G0ºG0º

29. May 201229 (4,∞ ) (1, ∞ ) (10, ∞ ) (5, ∞ ) (3, ∞ ) A: (3,2) B: (3,8) (1,8) (7,8) Critical Orientations correspond to slants of edges.

30. May 201230 Connectivity Graph G is constructed by stacking the connectivity graphs G θ corresponding to the critical orientations. Vertices of two distinct G θ are connected iff they are labeled with the same edge pair. Starting from G 0, G θ are added in increasing order of θ, thus creating a layered 3D graph. A paths from start to goal if exists can be found by a BFS algorithm.

31. May 201231 History