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
– Shakey – mobile, wireless,
path-planning, Hough
transform, camera vision,
English commands, logical
reasoning
Shakey video
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
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
Three-dimensional Space
Shortest path passes
through none of the
vertices
• 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
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
• 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)
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c9
c15
c11
c13
c12
Adjacency Graph
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c1
c9
c15
c11
c13
c12
Adjacency Graph
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c1
c3
c9
c15
c11
c13
c12
Adjacency Graph
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c2
c1
c3
c9
c15
c11
c13
c12
Adjacency Graph
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c9
c11
c13
c12
c4
c2
c15
c14
c7
c15
c5
c8
c1
c11
c10
c3
c6
c9
c13
c12
Path Planner
• Search in adjacency graph for path from start
cell to goal cell
• First, find nodes in path
Adjacency Graph
c14
c4
c5
c2
c7
c8
c1
c10
c3
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c4
c2
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c14
c7
c15
c5
c8
c1
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c10
c3
c6
c9
c13
c12
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
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c9
c11
c13
c12
c4
c2
c15
c14
c7
c15
c5
c8
c1
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c10
c3
c6
c9
c13
c12
What if goal were here?
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c4
c2
c15
c14
c7
c15
c5
c8
c1
c11
c10
c3
c6
c9
c13
c12
Trapezoidal Decomposition
• Shoot rays up and down from each vertex
until they enter a polygon
– Naïve approach O(n2) (n vertices times n edges)
c14
c4
c5
c2
c7
c8
c1
c10
c3
c6
c9
c15
c11
c13
c12
Other Exact Decompositions
• Triangular cell
• Optimal triangulation is
NP-hard (exponential in
vertices)