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

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

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

May 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?

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

May Configuration space Work space

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

May 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

May 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

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May 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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May Extreme Points and Directions

May How complex is Minkowski sum of two polygons?

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May 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 P Edges of P and R are labeled counterclockwise b c d d c a a b

May a b c d P -R a b c d

May b c d d c a a b a b c d

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

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

May 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!

May Cell Decomposition Minkowski sums A B C Cell decomposition Obstacles ∞ R Ladder

May 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.

May ∞ R A B C disappeared Ladder is rotated

May 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

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

May 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.

May History