PolyDepth: Real-Time Penetration Depth Computation Using Iterative Contact-Space Projection

Transcription

1 ACM Transactions on Graphics (ToG), Vol. 31, No. 1, Article 5, 2012 PolyDepth: Real-Time Penetration Depth Computation Using Iterative Contact-Space Projection Changsoo Je Ewha Womans University and Sogang University Min Tang, Youngeun Lee, Minkyoung Lee, and Young J. Kim, Ewha Womans University

2 Interpenetration Problem Modeling problem Numerical problem Random sampling Penetration Non-penetration 2

3 How to separate them? Penetration Non-penetration 3

4 Penetration Depth (PD) Penetration Depth Minimum translational distance to separate intersecting rigid objects 4

5 Applications Rigid Body Dynamics Point of impulse application [Tang et al. 2009] 5

6 Applications Retraction-based Motion Planning C-Obstacle C-Free Retract in-collision samples to free space [Zhang et al. 2008] 6

7 Applications Penalty-Based 6DOF Haptics Penalty Forces [Kim et al. 03] 7

8 Computational Complexity of PD Exact PD: O n 6 for n faces A B Minkowski Sums (Translational Contact Space) A B 8

9 Main Contribution The first realtime PD algorithm for general polygonal models Source codes available: google polydepth 9

10 Tight upper bound Main Contribution Single global or multiple local solutions Iterative optimization using in/out projections Efficient initialization strategies 10

11 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 11

12 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 12

13 Previous Work Convex Polytopes [Cameron and Culley 1986], [Dobkin et al. 1993], [Agarwal et al. 2000], [Cameron 1997], [Bergen 2001], [Kim et al. 2004] Nonconvex Polyhedra [Kim et al. 2002], [Redon and Lin 2006], [Lien 2008, 2009], [Hachenberger 2009], [Fisher and Lin 2001], [Hoff et al. 2002], [Sud et al. 2006], [Tang et al. 2009], [Allard et al. 2010] Generalized Penetration Depth [Zhang et al. 2007], [Nawratil et al. 2009], [Ong and Gilbert 1996], [Ong 1993], [Weller and Zachmann 2009] 13

14 Previous Work - Limitations Too slow for interactive applications Non-Euclidean Lower bound solution - Does not guarantee full separation 14

15 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 15

16 Minkowski Sum Minkowski sums for two sets A and B A B = A B = a + b a A, b B a b a A, b B 16

17 Minkowski Sum A B B A 17

18 PD and Minkowski Sum A B o A B 18

19 PD and Minkowski Sum A B o B o A PD PD 19

20 PD Estimation A B o 20

21 PD Estimation A B o Penetration depth 21

22 PolyDepth Algorithm Local optimization - Contact space localization Iterative optimization - Repeated in- and out-projections 22

23 Iterative Optimization o A B contact space 23

24 Iterative Optimization q f o A B contact space 24

25 Iterative Optimization Out-Projection q f q 0 A B o 25

26 Iterative Optimization q 0 A B o 26

27 Iterative Optimization q 1 In-Projection A B o 27

28 Iterative Optimization q 1 Out-Projection q 2 A B o 28

29 Iterative Optimization q 2 A B o 29

30 Iterative Optimization q 3 In-Projection PD o A B 30

31 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 31

32 Out-Projection q f A B o 32

33 Out-Projection Out-Projection q 0 q f A B o 33

34 Continuous Collision Detection (CCD) Find the first time of contact (ToC) τ 34

35 Translational CCD Find the first time of contact (ToC) τ 35

36 Translational CCD A v A τ d v B τ = d v A, B v d v : Minimal directional distance along v 36

37 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 37

38 Contact-Space Localization q f A B q 0 o 38

39 Contact-Space Localization A B q 0 o 39

40 Contact-Space Localization A B q 0 Jq = c 40

41 In-Projection A B q 1 In-Projection o 41

42 In-Projection A B Minimize q 2 q subject to Jq c o 42

43 In-Projection Linear Complementarity Problem q 1 4 JJT λ + c 0 λ JJT λ + c T λ = 0 o 43

44 In-Projection q JJ T λ = 4c o 44

45 In-Projection q Projected Gauss-Seidel method o λ k+1 = D + L 1 4c Uλ k where D + L + U = JJ T 45

46 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 46

47 Collision-Free Configuration for Out-Projection q f A B o 47

48 Initialization Find a collision-free configuration as close as possible to the relative origin Strategies: - Centroid difference - Maximally clear configuration - Motion coherence - Random configuration - Sampling-based search 48

49 Centroid Difference q s q A q B o B o A 49

50 Centroid Difference Initial collision-free Input in-collision In- & outprojections 50

51 Maximally Clear Configuration Equidistant Voronoi boundary Approximate the maximally clear configurations 51

52 Maximally Clear Configuration 1. AABB 52

53 Maximally Clear Configuration 2. Grid positions 53

54 Maximally Clear Configuration 3. Remove the points with small distances xy 54

55 Maximally Clear Configuration 4. Remove the closer points along xy-directions xy 55

56 Maximally Clear Configuration z 5. Remove the grids on the boundary of AABB 56

57 Maximally Clear Configuration z 6. Remove the closer points along z-direction 57

58 Maximally Clear Configuration 58

59 Maximally Clear Configuration PD Initial collision-free Input in-collision 59

60 Sampling-Based Search q 0 f Collision-free q 1 f A B o Collision Collision 60

61 Maximally Clear Configurations and Sampling-Based Search Initial collisionfree PD Input in-collision 61

62 Maximally Clear Configurations and Sampling-Based Search Input in-collision Initial collision-free PD 62

63 Motion Coherence in Dynamics Many collision-free configurations are found 63

64 Motion Coherence PD Initial collision-free Outprojections Input in-collision 64

65 Contents Previous work Preliminaries Out-projection In-projection Initialization Results Conclusion 65

66 Benchmarking Models 10K 1K 40K 1K 1.33K 174K 105K 0.54K 0.94K 152K 1.34K 2K 3K 66

67 Performance with Random Configurations 67

68 Performance with Random Configurations top : no. of contacts mid : no. of iterations bot : timing 68

69 Performance with Random Configurations Model Time (msec) No. of Contacts No. of Iterations Torusknot Buddha Bunny Dragon

70 Comparison to Lien s Method [Lien 2009] 2,000 Times Faster top : PD for mid : timing for bot : PD for 70

71 Average PD Errors Model Mean error (%) Median error (%) Bunny Distorted- Torus

72 Predefined Path Scenario 72

73 Dragons with Predefined Path top : no. of contacts mid : no. of iterations bot : timing 73

74 Performance on Predefined Paths Models Time (msec) No. of Contacts No. of Iterations Spoon and Cup Buddha Dragon

75 Realtime PD Results Spoon: 1.3K triangles Cup: 8.4K triangles Buddha 1: 10K triangles Buddha 2: 20K triangles 75

76 Applications Rigid Body Dynamics Multiple contacts To stabilize dynamics 76

77 Local PD Computation Computed for each contact region Derived from the global PD Corresponding to contact feature pairs that are resolved as a result of the global PD 77

78 Local PD Computation Two objects are interpenetrated. 78

79 Local PD Computation d d: global PD 79

80 Local PD Computation d n i : contact normals n 1 n 2 80

81 Local PD Computation d i-th local PD: d i =(d n i )n i d 1 d 2 81

82 Dynamics Simulations 82

83 Performance in the Dynamics Scenario Models Global PD (msec) Local PD (msec) Total (msec) Torus Bunny Golf-club Santa

84 Penalty-Based Haptic Rendering Box and Box Spoon and Cup 84

85 Realtime Rigid Body Dynamics 1 Plane: 12 triangles 7 Bunnies: 40K each 3 Dragons: 174K each Total: 802K Performance: 10~50 msec 85

86 Conclusion Interactive PD algorithm for complicated polygon-soup models Several schemes for selecting an initial collision-free configuration Method of computing a local PD for realistic dynamics simulation 86

87 Limitations Highly complex models may require a large number of iterations Approximates an upper bound of PD Depends on initial collision-free configurations 87

88 Future Work Extension to n-body PD problems Extension to generalized PD problem Applying to haptic rendering and robot motion planning 88

89 Acknowledgements Graphics group, Ewha Stephane Redon, INRIA Jyh-Ming Lien, GMU Anonymous reviewers National research foundation of Korea IT R&D Program of MKE/MCST in Korea 89

90 Thank You! C. Je et al. PolyDepth: Real-time Penetration Depth Computation. ACM Transactions on Graphics (ToG), Vol. 31, No. 1, Article 5. Presented in SIGGRAPH