Exact Collision Detection Recent Developments and Applications Dan - PowerPoint PPT Presentation

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 3D Minkowski sums the Voronoi-visibility complex future directions MOVIE, Toulouse 2005 Exact Collision Detection 2

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[Snoeyink-Stolfi ’93] MOVIE, Toulouse 2005 Exact Collision Detection 4

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) MOVIE, Toulouse 2005 Exact Collision Detection 5

Infrastructure significant progress in certified implementation of geometric algorithms the C GAL project and library C GAL : computational geometry algorithms library the C GAL arrangement package exact number types: L EDA , C ORE , C GAL MOVIE, Toulouse 2005 Exact Collision Detection 6

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Earlier results I: Exact 2D Minkowski sums [Flato-H, 2000] MOVIE, Toulouse 2005 Exact Collision Detection 8

� 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 MOVIE, Toulouse 2005 Exact Collision Detection 9

Earlier results II: Hybrid motion planning for 2 discs [Hirsch-H 2002] MOVIE, Toulouse 2005 Exact Collision Detection 10

Outline, recent results background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions MOVIE, Toulouse 2005 Exact Collision Detection 11

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 MOVIE, Toulouse 2005 Exact Collision Detection 12

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 MOVIE, Toulouse 2005 Exact Collision Detection 13

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 ξ (−2.4,6) (0.6,6) (−0.93,6) (0.296,2.96) x (−3,0) (0,0) MOVIE, Toulouse 2005 Exact Collision Detection 14

✂ ✁ 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 MOVIE, Toulouse 2005 Exact Collision Detection 15

Outline, recent results background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions MOVIE, Toulouse 2005 Exact Collision Detection 16

Motivation: Collision detection and proximity queries [Fogel-H 04] MOVIE, Toulouse 2005 Exact Collision Detection 17

✄ ☎ ✆ ✝ ☎ The Cubical Gaussian Map The Cubical Gaussian Map (CGM) of 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 MOVIE, Toulouse 2005 Exact Collision Detection 18

A dioctagonal pyramid and the Minkowski sum of 2 orthogonal dioctagonal pyramids The primal The CGM The CGM unfolded MOVIE, Toulouse 2005 Exact Collision Detection 19

An icosahedron and the Minkowski sum of 2 identical icosahedrons The primal The CGM The CGM unfolded MOVIE, Toulouse 2005 Exact Collision Detection 20

A geodesic sphere and the Minkowski sum of 2 slightly rotated geodesic spheres The primal The CGM The CGM unfolded MOVIE, Toulouse 2005 Exact Collision Detection 21

Experimental Results Time consumption of Minkowski-sum computation Minkowski Sum Object 1 Object 2 CH CGMO Primal Dual V E F V HE F Icosahedron Icosahedron 12 30 20 72 192 36 0.13 0.03 Dioctagonal Orthogonal 162 352 192 344 1136 236 0.53 0.28 Pyramid Dioctagonal Pyramid Pentagonal Truncated 527 964 439 719 2744 665 16.76 1.21 Hexeconta- Icosidodec- hedron ahedron Geodesic Rotated 814 1952 1146 1530 5060 1012 134.35 2.53 Sphere Geodesic level 4 Sphere level 4 MOVIE, Toulouse 2005 Exact Collision Detection 22

Outline, recent results background, motivation, earlier results multi-axis NC machining exact 3D Minkowski sums the Voronoi-visibility complex future directions MOVIE, Toulouse 2005 Exact Collision Detection 23

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 obstacle smooth — not containing sharp turns MOVIE, Toulouse 2005 Exact Collision Detection 24

Take I: Visibility diagrams used to plan shortest paths disadvantage: outputs semi-free paths (not having any clearance) MOVIE, Toulouse 2005 Exact Collision Detection 25

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; outputs non-smooth paths MOVIE, Toulouse 2005 Exact Collision Detection 26

✡ ☛ ✞✠✟ New construct: The VV diagram outputs shortest paths with a preferred clearance can be used to balance between path length and clearance as desired computed exactly on top of C GAL with fast path-query time MOVIE, Toulouse 2005 Exact Collision Detection 27

✔ ✏ ✎ ✌ ☞ ✓ ☛ ✓ ✒ ☛ ☛ ✛ ✑ ✏ ☛ ✓ ✎ ✎ ✕ ✜ 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 MOVIE, Toulouse 2005 Exact Collision Detection 28

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 MOVIE, Toulouse 2005 Exact Collision Detection 29

THE END MOVIE, Toulouse 2005 Exact Collision Detection 30