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Exact Collision Detection Recent Developments and Applications Dan Halperin danha@tau.ac.il Tel Aviv University MOVIE, Toulouse 2005 Exact Collision Detection 1 Outline background, motivation, earlier results multi-axis NC machining exact

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Exact Collision Detection

Recent Developments and Applications

Dan Halperin

danha@tau.ac.il

Tel Aviv University

MOVIE, Toulouse 2005 Exact Collision Detection 1

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Outline

background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions

MOVIE, Toulouse 2005 Exact Collision Detection 2

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MOVIE, Toulouse 2005 Exact Collision Detection 3

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[Snoeyink-Stolfi ’93]

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Motivation

exact collision detection, exact motion planning intricate tasks, tight quarters, assembly planning existing motion-planning systems are often unpredictable and incomplete (though in some restricted models they are complete)

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Infrastructure

significant progress in certified implementation of geometric algorithms the CGAL project and library

CGAL: computational geometry algorithms library

the CGAL arrangement package exact number types: LEDA, CORE, CGAL

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Earlier results I: Exact 2D Minkowski sums

[Flato-H, 2000]

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Is exactness a must? is it feasible?

exactness is a convenient way to certified geometric computing (certified computing with limited precision is typically hard) infeasible with many degrees of freedom

  • hybrid solutions

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Earlier results II: Hybrid motion planning for 2 discs

[Hirsch-H 2002]

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Outline, recent results

background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions

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Multi-axis NC machining

[Elber-Illushin-H-Wein 04] (1) collision detection: exact (2) continuous path verification: exact for translations (3) on-going, path correction: hybrid

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Exact collision detection

exploiting the rotational symmetry of the tool radial projection of the polyhedral toolpiece and machine, resulting in hyperbolic arcs and line segments comparing the tool profile with the lower envelope of the projected arcs

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Continuous path verification

the motion of the tool has 5 DOFs RT-decomposition: subpaths of pure translation or pure rotation collision detection along a pure subpath: comparing the silhouettes of 3D surfaces with the profile of the tool; translational silhouette computed exactly

x ξ

(0.6,6) (0,0) (−3,0) (−2.4,6) (0.296,2.96) (−0.93,6)

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Experimental results

Model Model Total Average Name Size

✁ ✂

Time Query Time Wineglass 2700 0.08 6562 101 0.0153 0.5 5768 707 0.1226 Teapot 12600 0.08 29952 2744 0.0916 0.05 55136 5515 0.1000 Turbine 40046 0.05 2430 255 0.1048 0.04 10900 733 0.0672 time in seconds on a 2.4 GHz machine, using fp

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Outline, recent results

background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions

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Motivation: Collision detection and proximity queries

[Fogel-H 04]

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The Cubical Gaussian Map

The Cubical Gaussian Map (CGM)

✄
  • f a polytope
☎

in

✆ ✝

is a set-valued function from

☎

to the six faces of the unit cube whose edges are parallel to the major axes and are of length two. A Tetrahedron The primal The CGM The CGM unfolded

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A dioctagonal pyramid and the Minkowski sum of 2 orthogonal dioctagonal pyramids

The primal The CGM The CGM unfolded

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An icosahedron and the Minkowski sum of 2 identical icosahedrons

The primal The CGM The CGM unfolded

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A geodesic sphere and the Minkowski sum

  • f 2 slightly rotated geodesic spheres

The primal The CGM The CGM unfolded

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Experimental Results

Time consumption of Minkowski-sum computation

Object 1 Object 2 Minkowski Sum CH CGMO Primal Dual V E F V HE F Icosahedron Icosahedron 12 30 20 72 192 36 0.13 0.03 Dioctagonal Pyramid Orthogonal Dioctagonal Pyramid 162 352 192 344 1136 236 0.53 0.28 Pentagonal Hexeconta- hedron Truncated Icosidodec- ahedron 527 964 439 719 2744 665 16.76 1.21 Geodesic Sphere level 4 Rotated Geodesic Sphere level 4 814 1952 1146 1530 5060 1012 134.35 2.53

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Outline, recent results

background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions

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Motivation

planning “natural-looking” collision-free motion paths for a robot or a group of entities between given start and goal short — not containing unnecessary detours having some clearance — not getting too close to an

  • bstacle

smooth — not containing sharp turns

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Take I: Visibility diagrams

used to plan shortest paths disadvantage: outputs semi-free paths (not having any clearance)

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Take II: Voronoi diagrams

using the “retraction method” we compute the Voronoi diagram of the obstacles to obtain paths with maximal clearance disadvantage: does not compute the shortest path;

  • utputs non-smooth paths

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New construct: The VV

✞✠✟ ✡

diagram

  • utputs shortest paths with a preferred clearance
☛

can be used to balance between path length and clearance as desired computed exactly on top of CGAL with fast path-query time

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The VV complex

the VV

☞✍✌ ✎

diagram interpolates between the visibility diagram (

☛ ✏ ✑

) and the Voronoi diagram (

☛ ✏ ✒

) of a set of obstacles the VV complex encapsulates the VV

☞ ✌ ✎

diagrams for all

☛

values easily queried for a fixed

☛

— no need to recostruct the VV

☞✍✌ ✎

Diagram for polygons with a total of

✓

vertices computed in

✔ ✕ ✓ ✖✘✗✚✙ ✛ ✓ ✜

time

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Further work

path correction in multi-axis NC machining exploiting spatial and temporal coherence in 3D collision detection group-flow optimization using the Voronoi-visibility complex certified geometric primitives for motion planning

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THE END

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