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Visibility Graph Team 10 NakWon Lee, Dongwoo Kim

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Robot Motion Planning

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Configuration Space

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Shortest Paths

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Shortest path in road map is not a real shortest path because some arcs are btw nodes that are far apart, whereas others are btw nodes that are close to each other. It is just minimum number of hop. To improve this problem, give each arc a weight corresponding to the Euclidean length, and use graph search algorithm that find the shortest path in a weighted graph. Dijkstra's algorithm But, it is still not a shortest path.

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Shortest Paths Think of this path as an elastic rubber band. Fix the endpoints at the start and goal position. Try to tighten the rubber band. It will be stopped by the obstacles. The new path will follow parts of the obstacle boundaries and straight line segments through open space.

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Shortest Paths

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short cut

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Shortest Paths

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Visibility Graph Definition: The visibility graph of s and t and the obstacle set is a graph whose vertices are s and t the obstacle vertices, and vertices v and w are joined by an edge if v and w are either mutually visible or if (v, w) is an edge of some obstacle.

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Visibility Graph

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Visible Vertices

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Visible Definition Two points p and q are mutually visible if the open line segment joining them doesn't intersect the interior of any obstacle.

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Search Tree

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Computing the Visibility Graph

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The question is what are the significant event points, and what happens with each event?

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Computing the Visibility Graph

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event

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Computing the Visibility Graph

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Dual Arrangement Dual of point D Dual of point C x-coordinate is the slope of dual segment

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Find Events

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Angular order By using topological plane sweep, event do not need to be sorted.

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Topological Plane Sweep

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Upper horizon tree Lower horizon tree

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Topological Plane Sweep

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What Happens at Each Event

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Possible scenarios

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What Happens at Each Event

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Thank You Q&A

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UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 O’Rourke Chapter 8 Motion Planning.

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