Visibility Graph Team 10 NakWon Lee, Dongwoo Kim

Robot Motion Planning

Configuration Space

Shortest Paths

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.

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.

Shortest Paths

short cut

Shortest Paths

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.

Visibility Graph

Visible Vertices

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.

Search Tree

Computing the Visibility Graph

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

Computing the Visibility Graph

event

Computing the Visibility Graph

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

Find Events

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

Topological Plane Sweep

Upper horizon tree Lower horizon tree

Topological Plane Sweep

What Happens at Each Event

Possible scenarios

What Happens at Each Event

Thank You Q&A