visibility graphs and cell decomposition by david johnson
TRANSCRIPT
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
• Start with geometry of robot and obstacles, R and O
• Compute the Minkowski difference of O – R
• Compute visibility graph in C-space
• Search graph for shortest path
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
tangent segments
Eliminate concave obstacle vertices(line would continue on into obstacle)
Reduced VG
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
Roadmaps and Coverage
• 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 and Coverage
• Roadmaps not so good for coverage of free space– Vacuum robots– Minesweeper robots– Farming robots
• Try to characterize the free space
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
• Adjacency graph:– 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
Trapezoidal Decomposition
• Two-dimensional cells that are shaped like trapezoids (plus special case triangles)
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Adjacency Graph
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Path Planner
• Search in adjacency graph for path from start cell to goal cell
• First, find nodes in path
Adjacency Graph
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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
Adjacency Graph
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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)
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