The Visibility Voronoi Complex and Its Applications

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The Visibility Voronoi Complex and Its
Applications

• Ron Wein
• Jur P. van den Berg (U. Utrecht)
• Dan Halperin

Nachsholim May 2005
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Motivation
Given a robot translating on the plane amidst a
set P1, , Pm of configuration-space obstacles,
and given start and goal configurations s and g,
we wish to plan a natural looking
collision-free motion path for the robot between
s and g.

• By natural looking we mean the path should be
• Short not containing unnecessary long detours.
• Having some clearance not getting to close to
  an obstacle.
• Smooth not containing sharp turns (C1-smooth).

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Application
Kamphuis and Overmars (2004)
Motion planning for coherent groups of entities
(computing a backbone path).
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Visibility Graphs
Lozano-Pérez (1979)
Used to plan shortest paths. Constructed in O(n2
log n) time, where n is the total number of
vertices. Output-sensitive O(n log n S)
algorithms also exist. Query time is O(n log n
S), using Dijkstras algorithm.
Resulting paths have no clearance
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The Retraction Method
ÓDúnglaing and Yap (1985), Rohnert (1991)
Construct the Voronoi diagram of the obstacles in
O(n log n). Query time is O(log n k).
Paths may be too long and may contain sharp turns
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The VV(c)-Diagram
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Properties of the VV(c)-Diagram

• Outputs shortest paths which keep the preferred
  amount of clearance from the obstacles, where
  possible.
• Paths are smooth.
• In case of narrow passages (between chain
  points), we get a path with the maximal possible
  clearance (locally).

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The VV-Complex
The VV(c)-diagram interpolates between the
visibility graph and the Voronoi diagram of the
obstacles.

• The VV-complex encodes all VV(c)-diagrams for all
  c values.
• Easily queried for any given c, without the need
  to construct the VV(c)-diagram.
• Constructed in O(n2 log n) time, handling ?(n2)
  events.

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The VV-Complex (II)
Suppose the clearance value grows from 0 to ?. We
can associate with each edge e we encounter a
validity rangeR(e) cmin(e), cmax(e) of
c-values for which this edge is valid. The
VV-complex of the scene of polygonal obstacles
P1, , Pm is therefore

• The Voronoi diagram V of the obstacles.
• A set T of interval trees For each obstacle
  vertex u (and for each chain point p), T(u) (or
  T(p)) store the incident edge to u (to p) indexed
  by their validity range.

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Bitangents to Dilated Vertices
v
uvll
uvrl
uvlr
u
uvrr
For each obstacle vertex u we keep two circular
lists Ll(u) and Lr(u) of left and right
bitangents, sorted by their slopes.
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Computing the VV-Complex (Initialization)

• Compute the visibility graph of the obstacles P1,
  , Pm.
• Examine each bitangent edge in the visibility
  graph and assign 0 to be the minimum value of its
  validity range.
• Initialize an event queue Q , construct Ll(u) and
  Lr(u) for each obstacle vertex u, and compute the
  initial visibility events based on adjacencies in
  these lists.
• Compute the Voronoi diagram of P1, , Pm.
• For each Voronoi arc a that contains the minimum
  clearance value cmin of its chain ?a, insert the
  chain event ?cmin, a? into Q .

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Visibility Events

• Occur when two edges uv and uw become equally
  sloped.
• We assign a maximal value for the validity range
  of the blocked edges.
• Some edges may become valid, and we assign a
  minimal value for their validity ranges.
• There are ?(n2) visibility events, each takes
  O(log n) to handle.

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Chain Events

• Occur when a chain start appearing in the
  VV(c)-diagram.We create two chain points
  associated with this chain.
• There are ?(n) chain events, each takes O(n log
  n) to handle.
• The motion of the chain points along the chain
  causestangency events and endpoint events.
• All these events are handled at O(n2 log n) time
  in total.

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Life-Cycle of an Edge
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Querying the VV-Complex
Given a start configuration s, a goal
configuration g and a preferred clearance value
c

• For each Voronoi chain compute the chain points
  (two at most) that correspond to the given
  c-value.
• Perform radial sweep from s and from g and obtain
  their incident visibility edges.
• Execute Dijkstras algorithm from s. The graph is
  implicitly maintained, as we obtain the incident
  edges of each vertex x we encounter from T(x). We
  do this until reaching g.

The total query time is O(n log n k), where k
is the number of edges encountered during the
search.
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Implementation and Experiments
We have implemented a CGAL-based application that
can robustly construct the VV(c)-diagram for a
given c-value. We employ

• CGALs segment Voronoi diagram package
  (Karavelas).
• CGALs arrangement package with the conic-arc
  traits.
• The GMP and CORE number types.

Diagram construction takes 360 seconds. Query
time is 0.020.1 seconds (compared with 0.51
seconds that a smoothing phase would consume).
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Thank you!