Visibility Graphs and Cell Decomposition

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# Visibility Graphs and Cell Decomposition - PowerPoint PPT Presentation

Visibility Graphs and Cell Decomposition. By David Johnson. Shakey the Robot. Built at SRI Late 1960’s For robotics, the equivalent of Xerox PARC’s Alto computer Alto – mouse, GUI, network, laser printer, WYSIWYG, multiplayer computer game

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## Visibility Graphs and Cell Decomposition

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### Visibility GraphsandCell Decomposition

By

David Johnson

Shakey the Robot
• Built at SRI
• Late 1960’s
• For robotics, the equivalent of Xerox PARC’s Alto computer
• Alto – mouse, GUI, network, laser printer, WYSIWYG, multiplayer computer game
• Shakey – mobile, wireless, path-planning, Hough transform, camera vision, English commands, logical reasoning
Shakey path planning
• Represent the world as a hierarchical grid
• Full
• Partially-full
• Empty
• Unknown
• Compute nodes at corners of objects
• Find shortest path through nodes – A*
Shakey used two good ideas
• A*
• Putting sub-goals on corners of vertices
• This has been generalized into the idea of visibility graphs.
Visibility Graphs
• Define undirected graph VG(N,L)
• V = all vertices of obstacles
• N = V union (Start,Goal)
• L = all links (ni,nj) such that there is no overlap with any obstacle. Polygon edge doesn’t count as overlapping.
Reusing Visibility Graphs
• Add new visibility edges for new start/goal points
• The rest is unchanged
• Creates a roadmap to follow
Visibility Graph in Motion Planning
• Compute the Minkowski difference of O – R
• Compute visibility graph in C-space
• Search graph for shortest path
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Computing the Visibility Graph
• Brute force
• Check every possible edge against all polygon edges
Special Cases
• Do include polygon edges that don’t intersect other polygons
• Don’t include edges that cross the interior of any polygon
• Minkowski difference of original obstacles may overlap
Reduced VG

tangent segments

• Eliminate concave obstacle vertices
• (line would continue on into obstacle)
Generalized

tangency point

Shortest path passes

through none of the vertices

Three-dimensional Space
• Original paper split up long line segments so there were lots of vertices to work with
• Computing the shortest collision-free path in a general polyhedral space is NP-hard
• Exponential in dimension
• Visibility Graphs make a roadmap through space
• Roadmaps not so good for coverage of free space
• What kind of robot needs to cover C-free?
• Roadmaps not so good for coverage of free space
• Vacuum robots
• Minesweeper robots
• Farming robots
• Try to characterize the free space
Cell Decomposition
• Representation of the free space using simple regions called cells

A cell

Exact Cell Decomposition
• Exact Cell Decomposition
• Decompose all free space into cells

Exact

Approximate

Coverage
• Cell decomposition can be used to achieve coverage
• Path that passes an end effector over all points in a free space
• Cell has simple structure
• Cell can be covered with simple motions
• Coverage is achieved by walking through the cells
Cell Decomposition
• Two cells are adjacent if they share a common boundary
• Node correspond to a cell
• Edge connects nodes of adjacent cells
Path Planning
• Path Planning in two steps:
• Planner determines cells that contain the start and goal
• Planner searches for a path within adjacency graph

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Trapezoidal Decomposition
• Two-dimensional cells that are shaped like trapezoids(plus special case triangles)

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Path Planner
• Search in adjacency graph for path from start cell to goal cell
• First, find nodes in path

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Creating a Path
• Trapezoid is a convex set
• Any two points on the boundary of a trapezoidal cell can be connected by a straight line segment that does not intersect any obstacle
• Path is constructed by connecting midpoint of adjacency edges

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What if goal were here?

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Trapezoidal Decomposition
• Shoot rays up and down from each vertex until they enter a polygon
• Naïve approach O(n2) (n vertices times n edges)
Other Exact Decompositions
• Triangular cell
• Optimal triangulation is NP-hard (exponential in vertices)