Visibility Graph and Voronoi Diagram

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Visibility Graph and Voronoi Diagram

• CS 326 Lecture 5, Part I

2
Common Framework

• Roadmap approach
• Limited to low-dimensional C
• essentially 2D polygonal regions, objects and
  obstacles.
• Good for Multi-queries
• Complete algorithms

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Visibility Graphs

• non-directed graph G
• nodes initial and goal configurations
  vertices of C-obstacles
• edges straight-line segments connecting two
  nodes that dont go through obstacles

qqoal
qinit

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Why it works

• Proposition
• ? a semi-free path between qinit and qqoal
• ?
• ? a simple polygonal line t lying in cl(Cfree)
  whose endpoints are qinit and qqoal, and such
  that ts vertices are vertices of CB

• Sufficient to consider only polygonal lines
  running via vertices of CB, and if such path
  exists, G contains the shortest one.

Visibility Graph
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The Algorithm

• Construct a visibility graph G
• Search G for a path from qinit to qqoal
• If a path is found, return it otherwise,
  indicate failure
• Construction most expensive
• - naively O(n3)
• - sweep-line algorithm renders it O(n2 log n)
• - O(n2) proposed.

Visibility Graph
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Reduced Visibility Graphs

• G without non-tangent segments and concave
  vertices.
• Tangent segment a segment tangent to CB at both
  nodes

qqoal
qinit
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Reduced Visibility Graphs
qqoal
qinit

• Algorithm O( n log n) possible.

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Voronoi Diagram

• A roadmap method based on retraction
• a continuous function of Cfree defined onto a
  1D subset of itself.
• A realization of retraction in 2D C with
  polygonal obstacles
• consists of curves in Cfree that are
  equidistant from two or more points in ?Cfree

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Voronoi Diagram II

• When cl(Cfree) is a bounded polygonal region,
  the diagram consists of straight and parabolic
  arcs
• straight set of configuration closest to the
  same pair of edges or vertices
• parabolic same pair of an edge and a vertex

• Maximize clearance between the robot and
  obstacles

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Why it works

• Let p Cfree ? R be a connectivity-preserving
  retraction.
• Proposition
• ? a free path between qinit and qqoal
• ?
• ? a path in roadmap R between p(qinit) and
  p(qqoal).

q
p(q)
Voronoi Diagram
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The Algorithm

• Compute Vor(Cfree)
• Compute p(qinit) and p(qqoal) find the arcs
  containing them
• Search and return a path between them if exists
  otherwise, return failure
• Construction most expensive O(n log n)
• Other steps O(n)

Voronoi Diagram