Visibility Graphsand

Cell Decomposition

ByDavid 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

tangent segmentsEliminate concave obstacle vertices(line would continue on into obstacle)

Reduced VG

tangency point

Generalized

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

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)

c11c1

c2

c4

c3

c6

c5 c8

c7

c10

c9c12

c13

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c15

Adjacency Graph

c1

c11c1

c2

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c5 c8

c7

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Adjacency Graph

c1

c3

c11c1

c2

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c5 c8

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Adjacency Graph

c1

c2

c3

c11c1

c2

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c5 c8

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Adjacency Graph

c1 c10

c2

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c4 c5

c6

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c11c1

c2

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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

c1 c10

c2

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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

c1 c10

c2

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c4 c5

c6

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c9

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c15

c11c1

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What if goal were here?

c1 c10

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c11c1

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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)

c11c1

c2

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c5 c8

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Other Exact Decompositions

• Triangular cell• Optimal triangulation is

NP-hard (exponential in vertices)