The Visibility Voronoi Complex and Its Applications

1
The Visibility Voronoi Complex and Its
Applications

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

January 2005
2
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).

3
Application
Motion planning for coherent groups of entities
(Kamphuis and Overmars, 2004).
4
Visibility Diagrams
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
5
The Retraction Method
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
6
Voronoi Arcs
There are O(n) arcs in the Voronoi diagram of a
set of polygons with a total of n vertices
u
u
w
v
t
u
w
v
v
Vertexedge
Edgeedge
Vertexvertex
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Voronoi Diagram of Polygons
Monotone chain
Voronoi vertex
Chain minimum
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The VV(c)-Diagram
Given a set of convex pairwise-disjoint polygonal
obstaclesP1, , Pm and a preferred clearance
value c, construct the following graph

• Compute the Voronoi diagram V of the obstacles.
• Dilate every obstacle Pi with a disc of radius c
  i.e. calculate the Minkowski sum Mi(c) Pi ?
  Bc.
• Compute the union M(c) ?i Mi(c).
• Compute the visibility graph G(c) of circular
  arcs and reflex vertices in the boundary of M(c).
• Compute M(c) ? V and merge the resulting portions
  of the Voronoi diagram with G(c) to form the
  final 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, we get a path with
  the maximal possible clearance (locally).

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The VV-Complex
Suppose the clearance value grows from 0 to ?. We
wish to associate with each edge e we encounter a
validity range R(e) cmin(e), cmax(e) of
c-values for which this edge is valid. Having
done so, we define the VV-complex of the scene of
polygonal obstacles P1, , Pm that stores

• 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 , sorting the events
  by their increasing c-order.
• For each obstacle vertex u, initialize Ll(u) and
  Lr(u). For each pair of neighboring edges e1, e2
  in these lists compute a c-value c where these
  edges become equally sloped (if one exists), and
  insert the visibility event ?c, e1, e2? into Q .
• 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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Handling Visibility Events

• Assign c to be the maximal c-value of the
  validity range of the edges uv and vw.
• Remove the other event involving uv from Q and
  delete this edge from L(u). Examine the new
  adjacency created in this list and insert the
  corresponding visibility event into the queue.
• Repeat step 2 for the opposite edge vu.
• If the edges uw and vw do not have a minimal
  value assigned to their validity range yet,
  assign c to it.

v
w
u
v
w
u
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Handling Chain Events

• Initiate two chain points p1(?a) and p2(?a)
  associated with the Voronoi chain, which will
  move toward as endpoints z1 and z2,
  respectively.
• Let u be the obstacle vertex that induces a. For
  each outgoing edge e ux, compute a c-value c
  for which e becomes incident to one of the chain
  points pi(?a). Insert the tangency event ?c, e,
  pi(?a)? into the event queue.
• If a is a vertexvertex arc, repeat step 2 for
  the other vertex v.
• Insert the endpoint events ?c(z1), p1(?a), z1?
  and ?c(z2), p2(?a), z2? into Q .

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Tangency and Endpoint Events
Tangency event An edge e ux (x may be either
a dilated obstacle vertex or a chain point)
becomes tangent to Bc(u) at a chain point p(?).
We replace e by a reincarnate p(?)x. Endpoint
event A chain point p(?a) reached the endpoint
z of the arc a, and moves to the next arc in the
chain ?a. The associated visibility events need
to be recomputed.
When we have several event occurring at the same
c-value, we deal with endpoint events first, then
with visibility events, then we handle chain
events and finally tangency events.
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Life-Cycle of an Edge
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Complexity Analysis

• The initialization takes O(n2 log n) time.
• O(n2) visibility events, each takes O(log n) time
  to handle.
• O(n) chain events, each takes O(n log n) time to
  handle.
• Each chain event spawns as many as O(n) tangency
  events, so in total we have O(n2) tangency
  events, each one is handled in O(log n).
• O(n) endpoint events, each takes O(n log n) time
  to handle.

The total preprocessing time is O(n2 log n).
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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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Representation of the VV(c) Arcs
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Representation of the VV(c) Arcs (II)
In case all obstacle vertices have rational
coordinates and the squared clearance value is
also rational, the VV(c)-diagram can be
represented by conic arcs with rational
coefficients.