Exact Arrangements and Motion Planning

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Exact Arrangementsand Motion Planning

• Ron Wein
  July 2003

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Planar Arrangements
Given a collection C of curves in the plane, the
arrangement of A(C) is the subdivision of the
plane into vertices, edges and faces induced by
the curves in C.
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CGALs Arrangement Package(Ezra, Flato, Fogel,
Hanniel, Hirsch, Nechushtan)
CGALs arrangement package handles the topology
of planar arrangements and separates it from the
geometry of the curves

• It constructs the underlying DCEL and maintains
  the connection between the half-edges and their
  generating curves.
• The arrangement may either be constructed
  incrementally,or using a sweep-line algorithm.
• Point-location queries are supported.

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Robustness Issues
The package is designed to handle all sorts of
degenerate inputs, such as
Assuming correctness of the geometric methods.
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The Arrangement Traits Concept
The geometric predicates and constructions are
supplied by a so-called traits class, which
should

• Divide a curve to x-monotone sub-curves.
• Compare two sub-curves at a given x-coordinate.
• Compare two sub-curves next to their
  intersection.
• Determine if a given point is above, below or on
  a sub-curve.
• Compute the intersection points (or overlaps) of
  two curves.

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Existing Arrangement Traits
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More Robustness Issues
The traits classes implement robust predicate and
exact constructions (e.g. of intersection points)
using exact arithmetic and exact number types

• The segment traits and the polyline traits use
  rational numbers with unbounded integers.
• The conic traits class uses algebraic numbers
  (from the LEDA or the CORE libraries).
• Filters are massively used for speed-ups.

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Translating Polygonal Robot
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Translating Polygonal Robot(E. Flato, 2000)

• Divide the robot into convex polygons Q1, , Qm.
• For each obstacle P(k)
• Divide into convex polygons P1(k), , Pnk(k).
• Compute
• Compute the union of the C(k) polygons and
  construct the vertical decomposition into
  pseudo-trapezoids.
• Construct the connectivity graph for the
  pseudo-trapezoids.

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Translating Disc Robot
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Translating Disc Robot(S. Hirsch and E.
Leiserowitz, 2001)

• Infalte each obstacle P(k) by the radius of the
  robot and obtain C(k), whose boundary is a
  collection of line segment and circular arcs.
• Compute the union of the C(k) pseudo-polygons and
  construct the vertical decomposition into
  pseudo-trapezoids.
• Construct the connectivity graph for the
  pseudo-trapezoids.

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Hybrid Motion Planning Two Discs Moving among
Polygonal Obtacles(S. Hirsch and D. Halperin,
2002)
There are 4 degrees of freedom. The problem is
reduced to simpler planar motion planning
problems, then lifted back to ?4.
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The Configuration Space
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Configurations for Two Discs

• Let D1 and D2 denote the two disc robots.

• Compute Cfree(1) and Cfree(2), the free
  configuration for each robot (disregarding the
  other robot).

• Let c1(1),, cm1(1) and c1(2),, cm2(2) be
  the decomposition of Cfree(1) and Cfree(2) to
  pseudo-discs.

• Let us denote Cijci(1)?cj(2), and Cfree
  ?i,jCi,j

• Denote Cfree- ?i,jCi,j (ci(1)?D1) ?
  (cj(2)?D2) ?

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Hybrid Roadmap Construction

• Construct Cfree(1) and Cfree(2) exactly, and
  compute the vertical decompositions c1(1),,
  cm1(1) and c1(2),, cm2(2).

• In-cell connection For each cell Cijci(1)?cj(2)
  construct its connectivity graph Gij using a
  local planner.

• Inter-cell connection For each pair of adjacent
  cells Cij and Ckl, connect connected components
  of Gij and Gkl.

• Stitching For each pair Vi and Vj of connected
  components of the initial roadmap G, try to merge
  the two components into a single connected
  component.

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Features of HyMP

• Reliance on exact computation wherever possible.

• The explicit representation of Cfree allows us
  to provide disconnection proofs.

• Insensitivity to narrow passages (and sometimes
  even tight passages).

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Experimental Results
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Coordinating a Milling CutterComputing the Lower
Envelope of Line Segments and Hyperbolic
Arcs(Ongoing work with G. Elber and O. Ilusihin)
Motion planning for a CAD application.
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Motivation
A milling cutter rotating in an environment of
polyhedral surfaces, and should be moved without
touching those surfaces.
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Exact Solution

• Select a plane that contains the symmetry axis of
  the rotating cutter.

• Compute the lower envelope of the collection and
  check whether it intersects with the cutter.

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Implementation Details

• A new CGAL package was developed for computing
  the lower (and upper) envelope of a collection of
  curves.
• The package employs the divide-and-conquer
  algorithm for computing the lower envelope in O(n
  logn) time.
• However, no general position assumptions are
  made
• Curves may overlap.
• Vertical segments are supported, etc.
• The package also separates the topology for the
  geometry. It makes use of a traits class with the
  same predicates and methods as the one use by the
  Arrangement package.

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Preliminary Results
Using exact computation, the lower envelope of
500 hyperbolic arcs is computed in 8 seconds
(where about 50 curves form the lower envelope).