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Fast Penetration Depth Computation for Physically-based Animation Y. Kim, M. Otaduy, M. Lin and D.

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1 http://gamma.cs.unc.edu/PD Fast Penetration Depth Computation for Physically-based Animation http://gamma.cs.unc.edu/PD Y. Kim, M. Otaduy, M. Lin and D. Manocha Computer Science UNC – Chapel Hill Y. Kim, M. Otaduy, M. Lin and D. Manocha Computer Science UNC – Chapel Hill

2 http://gamma.cs.unc.edu/PD Proximity Queries and PD Collision Detection Separation Distance Penetration Depth d d

3 http://gamma.cs.unc.edu/PD Definition  Penetration Depth (PD)  Minimum translational distance to separate two intersecting objects  Penetration Depth (PD)  Minimum translational distance to separate two intersecting objects PD

4 http://gamma.cs.unc.edu/PD Motivation  Contact Handling in Rigid Body Simulation  Time stepping method »Check the contact at fixed intervals  Penalty-based method  Contact Handling in Rigid Body Simulation  Time stepping method »Check the contact at fixed intervals  Penalty-based method

5 http://gamma.cs.unc.edu/PD Motivation  Time stepping method  Estimate the time of collision (TOC)  Time stepping method  Estimate the time of collision (TOC) T = t 0 T = t 1 T = t 2 T = t ?

6 http://gamma.cs.unc.edu/PD Motivation  Time stepping method using PD 1.Compute the penetration depth 2.Estimate the TOC by interpolation in time domain  Time stepping method using PD 1.Compute the penetration depth 2.Estimate the TOC by interpolation in time domain T = t k T = t 1 T = t 2

7 http://gamma.cs.unc.edu/PD Motivation  Penalty-based Method T = t 0 T = t 1 T = t 2

8 http://gamma.cs.unc.edu/PD PD Application  Rigid body dynamic simulation  Robot motion planning for autonomous agents and animated characters  Haptic rendering  Tolerance verification for CAD models  Rigid body dynamic simulation  Robot motion planning for autonomous agents and animated characters  Haptic rendering  Tolerance verification for CAD models

9 http://gamma.cs.unc.edu/PD Previous Work  Convex polytopes  [Cameron ’86 ’97], [Dobkin et al. ’93], [Agarwal et al. ’00], [Bergen ’01], [Kim et al. ’02]  Local solutions for deformable models  [Susan and Lin ’01], [Hoff et al. ’01]  No solutions existed for non-convex models  Convex polytopes  [Cameron ’86 ’97], [Dobkin et al. ’93], [Agarwal et al. ’00], [Bergen ’01], [Kim et al. ’02]  Local solutions for deformable models  [Susan and Lin ’01], [Hoff et al. ’01]  No solutions existed for non-convex models

10 http://gamma.cs.unc.edu/PD Overview  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation

11 http://gamma.cs.unc.edu/PD Overview  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation

12 http://gamma.cs.unc.edu/PD Preliminaries  Local solutions might not have any relevance to a global solution Local PD solutions PD Global PD solution

13 http://gamma.cs.unc.edu/PD Preliminaries  Minkowski sum and PD  P  -Q = { p-q | p  P, q  Q }  PD := minimum distance between O Q-P and the surface of P  -Q  Minkowski sum and PD  P  -Q = { p-q | p  P, q  Q }  PD := minimum distance between O Q-P and the surface of P  -Q  P  -Q P O Q-P OPOP OQOQ Q

14 http://gamma.cs.unc.edu/PD Preliminaries  Minkowski sum and PD  P  -Q = { p-q | p  P, q  Q }  PD := minimum distance between O Q-P and the surface of P  -Q  Minkowski sum and PD  P  -Q = { p-q | p  P, q  Q }  PD := minimum distance between O Q-P and the surface of P  -Q  P  -Q P O Q-P OPOP OQOQ Q

15 http://gamma.cs.unc.edu/PD Preliminaries  Decomposition property of Minkowski sum  If P = P 1  P 2, then P  Q = (P 1  Q)  (P 2  Q)  Computing Minkowski sum  Convex: O(n log(n)) »where n is the number of features  Non-Convex: O(n 6 ) computational complexity »In theory, use the convolution or the decomposition property »In practice, very hard to implement.  Decomposition property of Minkowski sum  If P = P 1  P 2, then P  Q = (P 1  Q)  (P 2  Q)  Computing Minkowski sum  Convex: O(n log(n)) »where n is the number of features  Non-Convex: O(n 6 ) computational complexity »In theory, use the convolution or the decomposition property »In practice, very hard to implement.

16 http://gamma.cs.unc.edu/PD Overview  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation

17 http://gamma.cs.unc.edu/PD PD Algorithm : A Hybrid Approach  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2

18 http://gamma.cs.unc.edu/PD PD Algorithm : A Hybrid Approach  P  Q  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  P  Q  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2

19 http://gamma.cs.unc.edu/PD PD Algorithm : A Hybrid Approach  P  Q = (P 1  Q)  (P 2  Q) P = P 1  P 2  where P = P 1  P 2  Precomputation: Decomposition  P  Q = (P 1  Q)  (P 2  Q) P = P 1  P 2  where P = P 1  P 2  Precomputation: Decomposition

20 http://gamma.cs.unc.edu/PD PD Algorithm : A Hybrid Approach P 1  QP 2  Q  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  Precomputation: Decomposition  Runtime:  Object Space: Pairwise Minkowski sum computation P 1  QP 2  Q  P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  Precomputation: Decomposition  Runtime:  Object Space: Pairwise Minkowski sum computation

21 http://gamma.cs.unc.edu/PD PD Algorithm : A Hybrid Approach   P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  Precomputation: Decomposition  Runtime:  Object Space: Pairwise Minkowski sum computation  Image Space: Union by graphics hardware   P  Q = (P 1  Q)  (P 2  Q)  where P = P 1  P 2  Precomputation: Decomposition  Runtime:  Object Space: Pairwise Minkowski sum computation  Image Space: Union by graphics hardware

22 http://gamma.cs.unc.edu/PD PD Computation Pipeline Pairwise Minkowski Sum (Object Space) Precomputation Run-time PD Query Closest Point Query (Image Space) Convex Decomposition

23 http://gamma.cs.unc.edu/PD Convex Surface Decomposition  [Ehmann and Lin ’01]  Decompose an object into a collection of convex surface patches  Compute the convex hull of each surface patch  [Ehmann and Lin ’01]  Decompose an object into a collection of convex surface patches  Compute the convex hull of each surface patch

24 http://gamma.cs.unc.edu/PD Pairwise Minkowski Sum  Algorithms  Convex hull property of convex Minkowski sum » P  Q = ConvHull{v i +v j | v i  V P, v j  V Q }, where P and Q are convex polytopes.  Topological sweep on Gauss map [Guibas ’87]  Incremental surface expansion [Rossignac ’92]  Algorithms  Convex hull property of convex Minkowski sum » P  Q = ConvHull{v i +v j | v i  V P, v j  V Q }, where P and Q are convex polytopes.  Topological sweep on Gauss map [Guibas ’87]  Incremental surface expansion [Rossignac ’92]

25 http://gamma.cs.unc.edu/PD Closest Point Query  Goal  Given a collection of convex Minkowski sums, compute the shortest distance from the origin to the surface of their union  An exact solution is computationally expensive  Approximation using graphics hardware  Goal  Given a collection of convex Minkowski sums, compute the shortest distance from the origin to the surface of their union  An exact solution is computationally expensive  Approximation using graphics hardware

26 http://gamma.cs.unc.edu/PD  Main Idea  Incrementally expand the current front of the boundary  Main Idea  Incrementally expand the current front of the boundary Closet Point Query

27 http://gamma.cs.unc.edu/PD Closest Point Query 1.Render front faces, and open up a window where z-value is less than the current front 2.Render back faces w/ z-greater-than test 3.Repeat the above m times, where m := # of obj’s 1.Render front faces, and open up a window where z-value is less than the current front 2.Render back faces w/ z-greater-than test 3.Repeat the above m times, where m := # of obj’s

28 http://gamma.cs.unc.edu/PD Closest Point Query 1.Render front faces, and open up a window where z-value is less than the current front 2.Render back faces w/ z-greater-than test 3.Repeat the above m times, where m := # of obj’s 1.Render front faces, and open up a window where z-value is less than the current front 2.Render back faces w/ z-greater-than test 3.Repeat the above m times, where m := # of obj’s

29 http://gamma.cs.unc.edu/PD Overview  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation

30 http://gamma.cs.unc.edu/PD Motivation  PD is shallow in practice.  Convex decomposition has O(n) convex pieces in practice.  Culling strategy is suitable and very effective.  PD is shallow in practice.  Convex decomposition has O(n) convex pieces in practice.  Culling strategy is suitable and very effective.

31 http://gamma.cs.unc.edu/PD Object Space Culling  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD u PD

32 http://gamma.cs.unc.edu/PD Object Space Culling  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD u PD PD

33 http://gamma.cs.unc.edu/PD Object Space Culling  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD  Basic Idea  If we know the upper bound on PD, u PD, we do not need to compute the Mink. sum of pairs whose Euclidean dist is more than u PD

34 http://gamma.cs.unc.edu/PD Image Space Culling  Rendering only once for the Minkowski sums containing the origin  Refine the upper bound for every view frustum  View frustum culling  Rendering only once for the Minkowski sums containing the origin  Refine the upper bound for every view frustum  View frustum culling

35 http://gamma.cs.unc.edu/PD  BVH Construction (Ehmann and Lin ’01) Hierarchical Culling  Hierarchical Culling

36 http://gamma.cs.unc.edu/PD  BVH Construction (Ehmann and Lin ’01) Hierarchical Culling  Hierarchical Culling

37 http://gamma.cs.unc.edu/PD Hierarchical Refinement Culling Pairwise Minkowski Sum BVH Construction Precomputation Run-time PD Query Refine the current PD estimate, and go to the next level of BVH Closest Point Query Convex Decomposition

38 http://gamma.cs.unc.edu/PD PD Benchmarks 0.3 sec (4 hr) 3.7 sec (4 hr) 1.9 sec (177 hr) 0.4 sec (7 min) With Accel. (Without Accel.) With Accel. (Without Accel.) Touching ToriInterlocked ToriInterlocked GratesTouching Alphabets

39 http://gamma.cs.unc.edu/PD PD Benchmarks ModelsTri Convex Pieces PD w/o Accel. PD w/ Accel Touching Tori2000674 hr0.3 sec Interlocked Tori2000674 hr3.7 sec Interlocked Grates444, 1134169, 409177 hr1.9 sec Touching Alphabets 144,15242, 437 min0.4 sec

40 http://gamma.cs.unc.edu/PD Hierarchical Culling Example LevelCull RatioMink SumHW QueryPD est 331.2%0.219 sec0.220 sec0.99 596.7%0.165 sec0.146 sec0.53 798.3 %1.014 sec1.992 sec0.50

41 http://gamma.cs.unc.edu/PD Implementation  SWIFT++ [Ehmann and Lin ’01]  QHULL  OpenGL for closest point query  SWIFT++ [Ehmann and Lin ’01]  QHULL  OpenGL for closest point query

42 http://gamma.cs.unc.edu/PD Implementation void DrawUnionOfConvex(ConvexObj *ConvexObjs, int NumConvexObjs) { glClearDepth(0); glClearStencil(0); glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT | GL_STENCIL_BUFFER_BIT); glEnable(GL_DEPTH_TEST); glEnable(GL_STENCIL_TEST); for (int i=0; i<NumConvexObjs; i++) for (int j=0; j<NumConvexObjs; j++) { glDepthMask(0); glColorMask(0,0,0,0); glDepthFunc(GL_LESS); glStencilFunc(GL_ALWAYS,1,1); glStencilOp(GL_KEEP,GL_REPLACE,GL_KEEP); ConvexObjs[j].DrawFrontFaces(); glDepthMask(1); glColorMask(1,1,1,1); glDepthFunc(GL_GREATER); glStencilFunc(GL_EQUAL,0,1); glStencilOp(GL_ZERO,GL_KEEP,GL_KEEP); ConvexObjs[j].DrawBackFaces(); } void DrawUnionOfConvex(ConvexObj *ConvexObjs, int NumConvexObjs) { glClearDepth(0); glClearStencil(0); glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT | GL_STENCIL_BUFFER_BIT); glEnable(GL_DEPTH_TEST); glEnable(GL_STENCIL_TEST); for (int i=0; i<NumConvexObjs; i++) for (int j=0; j<NumConvexObjs; j++) { glDepthMask(0); glColorMask(0,0,0,0); glDepthFunc(GL_LESS); glStencilFunc(GL_ALWAYS,1,1); glStencilOp(GL_KEEP,GL_REPLACE,GL_KEEP); ConvexObjs[j].DrawFrontFaces(); glDepthMask(1); glColorMask(1,1,1,1); glDepthFunc(GL_GREATER); glStencilFunc(GL_EQUAL,0,1); glStencilOp(GL_ZERO,GL_KEEP,GL_KEEP); ConvexObjs[j].DrawBackFaces(); }

43 http://gamma.cs.unc.edu/PD Accuracy of PD computation  Our algorithm computes an upper bound to PD  Image space computation determines the tightness of the upper bound  Pixel resolution  Z-buffer precision  In practice, with 256  256 pixel resolution, the algorithm rapidly converges to the PD  Our algorithm computes an upper bound to PD  Image space computation determines the tightness of the upper bound  Pixel resolution  Z-buffer precision  In practice, with 256  256 pixel resolution, the algorithm rapidly converges to the PD

44 http://gamma.cs.unc.edu/PD Overview  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation  Preliminaries  Minkowski sum-based Framework  Basic PD Algorithm: A Hybrid Approach  Acceleration Techniques  Object Space Culling  Hierarchical Refinement  Image Space Culling  Application to Rigid Body Dynamic Simulation

45 http://gamma.cs.unc.edu/PD  Interpenetration is often unavoidable in numerical simulations  Need for a consistent and accurate measure of PD  Penalty-based Method  F = (k d)n »d: PD, n: PD direction, k:stiffness constant  Interpenetration is often unavoidable in numerical simulations  Need for a consistent and accurate measure of PD  Penalty-based Method  F = (k d)n »d: PD, n: PD direction, k:stiffness constant Application to Rigid Body Simulation

46 http://gamma.cs.unc.edu/PD Application to Rigid Body Simulation  Time Stepping Method  Estimate the time of collision (TOC) »s: separation dist before interpenetration »d: PD, n: PD dir after interpenetration »v s : relative velocity of closest features »v d : relative velocity of PD features  Time Stepping Method  Estimate the time of collision (TOC) »s: separation dist before interpenetration »d: PD, n: PD dir after interpenetration »v s : relative velocity of closest features »v d : relative velocity of PD features

47 http://gamma.cs.unc.edu/PD Application to Rigid Body Simulation  x(t) : 1D distance function between closest features and PD features projected to PD direction  x(0) = s, x(T) = d  dx/dt(0) = v sn, dx/dt(T) = v dn  Compute the roots of x(t) = 0  x(t) : 1D distance function between closest features and PD features projected to PD direction  x(0) = s, x(T) = d  dx/dt(0) = v sn, dx/dt(T) = v dn  Compute the roots of x(t) = 0

48 http://gamma.cs.unc.edu/PD Rigid Body Simulation Demo Example 1 Average complexity: 250 triangles 60 Convex Pieces / Object Example 2 Average complexity: 250 triangles 200 letters and alphabets

49 http://gamma.cs.unc.edu/PD Rigid Body Simulation Demo

50 http://gamma.cs.unc.edu/PD Contributions  First practical PD algorithm using a hybrid approach  Acceleration techniques:  Object space culling  Image space culling  Hierarchical refinement  Application to rigid body simulation  First practical PD algorithm using a hybrid approach  Acceleration techniques:  Object space culling  Image space culling  Hierarchical refinement  Application to rigid body simulation

51 http://gamma.cs.unc.edu/PD Future Work  More optimizations on  Closest point query  Pairwise Minkowski sum  More performance comparisons  Accuracy and running time of PD algorithm  Other RBD simulation approaches  Extension to rotational PD  More optimizations on  Closest point query  Pairwise Minkowski sum  More performance comparisons  Accuracy and running time of PD algorithm  Other RBD simulation approaches  Extension to rotational PD

52 http://gamma.cs.unc.edu/PD Acknowledgement  Kenneth Hoff  Stephen Ehmann  ARO  DOE  NSF  ONR  Intel  Kenneth Hoff  Stephen Ehmann  ARO  DOE  NSF  ONR  Intel

53 http://gamma.cs.unc.edu/PD Thank you http://gamma.cs.unc.edu/PD

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