1. UNC at CHAPEL HILL & Ewha Womans University Generalized Penetration Depth Computation Liang-Jun Zhang Gokul Varadhan Dinesh Manocha Dept of Computer Sci. University of North Carolina Chapel Hill, USA Young J. Kim Dept of Computer Sci. Ewha Womans University Seoul, Korea http://gamma.cs.unc.edu/pdg ACM Solid and Physical Modeling Conference, 2006

2. UNC at CHAPEL HILL & Ewha Womans University 2 Distance Measure Separation → Euclidean distance Interpenetration → Penetration depth d d

3. UNC at CHAPEL HILL & Ewha Womans University 3 Translational Penetration Depth (PD t ) Minimum translational distance needed to separate objects

4. UNC at CHAPEL HILL & Ewha Womans University 4 Translational Penetration Depth (PD t ) d Minimum translational distance needed to separate objects

5. UNC at CHAPEL HILL & Ewha Womans University 5 Applications of PD t Physically-based animation 6DOF haptic rendering Robot motion planning Tolerance verification Time of Contact Force

6. UNC at CHAPEL HILL & Ewha Womans University 6 PD g No Rotational Motion in PD t PD t

7. UNC at CHAPEL HILL & Ewha Womans University 7 Generalized Penetration Depth PD g ♦ Taking into account both translational and rotational motion

8. UNC at CHAPEL HILL & Ewha Womans University 8 Main Contributions Definition of PD g ♦ New distance metric, D g ♦ Properties Practical algorithms for PD g ♦ Convex/convex ♦ Convex/convex complement ♦ Non-convex/non-convex

9. UNC at CHAPEL HILL & Ewha Womans University 9 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

10. UNC at CHAPEL HILL & Ewha Womans University 10 Previous Work Translational Penetration Depth Intersection depth ♦ [Dobkin et al. 93] Convex polytope ♦ [Agarwal et al. 00] ♦ Lower bound [van den Bergen01] ♦ Upper bound [Kim et al. 02b] Non-convex polyhedra ♦ Higher complexity: O(n 6 ) ♦ [Kim et al. 02a], [Redon et al. 05]

11. UNC at CHAPEL HILL & Ewha Womans University 11 Previous Work Generalized Penetration Depth No directly related work ♦ 6DOF configuration space: O(n 12 ) complexity Object Containment problem ♦ [Chazella 83], [Milenkovic 99], [Grinde and Cavalier 96], [Avniam and Boissonnat 89], [Agarwal et al 98] Rotational overlapping minimization ♦ [Milenkovic 98], [Milenkovic and Schmidl 01] ♦ Quadratic metric

12. UNC at CHAPEL HILL & Ewha Womans University 12 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

13. UNC at CHAPEL HILL & Ewha Womans University 13 Configuration Space (C-space) Workspace C-space X Y X Y θ q 0 = q 1 = A

14. UNC at CHAPEL HILL & Ewha Womans University 14 To measure the distance for one object at two different configurations [LaValle06,Amato00,Kuffner04] ♦ L p metrics (L 2, L 1, L ∞ ) ♦ Displacement metric Maximum displacement ♦ Our D g distance metric Distance Metric in C-space X Y q0q0 q1q1 A(q 0 ) X Y θ A(q 1 ) d

15. UNC at CHAPEL HILL & Ewha Womans University 15 Min over every path connecting q 0 and q 1 Max trajectory length for distinct points D g (q 0, q 1 ) = D g Metric in C-space X Y θ q1q1 q0q0 l1l1 l2l2 Motion Paths in C-Space Trajectory length A(q 0 ) A(q 1 ) D g (q 0,q 1 )

16. UNC at CHAPEL HILL & Ewha Womans University 16 Properties of D g metric Non-negativityD g (q 0,q 1 )≥0 Reflexivity D g (q 0,q 1 )=0 ⇔ q 0 =q 1 SymmetryD g (q 0,q 1 )=D g (q 1,q 0 ) Triangle Inequality D g (q 0,q 1 )+D g (q 1,q 2 )≥D g (q 0,q 2 )

17. UNC at CHAPEL HILL & Ewha Womans University 17 PD g definition The minimum D g distance over all possible collision- free configurations A B PD g

18. UNC at CHAPEL HILL & Ewha Womans University 18 PD t = Special Case of PD g PD t : only by translation d B A A(q)

19. UNC at CHAPEL HILL & Ewha Womans University 19 PD g (A,B) ≤ PD t (A,B) PD g PD t

20. UNC at CHAPEL HILL & Ewha Womans University 20 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

21. UNC at CHAPEL HILL & Ewha Womans University 21 PD g for Convex Objects Theorem PD g (A,B) = PD t (A,B) Pf) 1.In general, PD g (A,B) ≤ PD t (A,B) 2.Show that PD g (A,B) < PD t (A,B) is impossible for convex objects

22. UNC at CHAPEL HILL & Ewha Womans University 22 Corollary Known PD t algorithms directly applicable to computing PD g PD g (A,B) = PD g (B,A) for convex objects In general, PD g (A,B)≠PD g (B,A) for non- convex objects

23. UNC at CHAPEL HILL & Ewha Womans University 23 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

24. UNC at CHAPEL HILL & Ewha Womans University 24 Object containment ♦ Can Q contain P, when P is allowed to translate and rotate? PD g ♦ Consider as the container Q ♦ Harder because we need to find an optimal PD g vs Containment

25. UNC at CHAPEL HILL & Ewha Womans University 25 Object Containment Problem Complexities ♦ O(m 3 n 3 log(mn)) for 2D polygons ♦ O(mn 2 ) for 2D convex polygons Motivates us to consider PD g between convex and convex complement

26. UNC at CHAPEL HILL & Ewha Womans University 26 Algorithm Overview 1.Find a containment 2.Find a locally optimal containment A

27. UNC at CHAPEL HILL & Ewha Womans University 27 Step 1: Find a Containment Formulate as a linear programming problem Containment constraint: ♦ Each point on A is contained in a half space defined by each face in B

28. UNC at CHAPEL HILL & Ewha Womans University 28 Step 2: Find a locally optimal containment Another Linear programming Optimization objective ♦ D g (q 0, q 2 ) ≤D g (q 0, q 1 ) D g (q 0,q 1 ) D g (q 0,q 2 )

29. UNC at CHAPEL HILL & Ewha Womans University 29 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

30. UNC at CHAPEL HILL & Ewha Womans University 30 PD g for Convex/Non-convex Difficult to compute the exact PD g ♦ May need to compute 6DOF C-space (O(n 12 )) Lower bound algorithm Upper bound algorithm

31. UNC at CHAPEL HILL & Ewha Womans University 31 Lower Bound on PD g 1.Convex decomposition 2.Eliminate non-overlapping pairs 3.PD t for overlapping pairs 4.LB(PD g ) = Max over all PD t s PD t

32. UNC at CHAPEL HILL & Ewha Womans University 32 Upper Bounds on PD g 1.PD g (A,B) ≤ PD t (CHull(A), CHull(B)) 2.PD g (A,B) ≤ PD t (A, B) More difficult 3. Better bounds

33. UNC at CHAPEL HILL & Ewha Womans University 33 Separating planes (b) (c) Convex separator Non-convex separator Separating Planes and Separators

34. UNC at CHAPEL HILL & Ewha Womans University 34 Separating Planes PD t (Chull(A), Chull(B)) yields a upper bound

35. UNC at CHAPEL HILL & Ewha Womans University 35 Convex Separators 1.Enumerating convex separators S 2.For each convex separator S, use convex/convex complement for an upper bound 3.Min over all these upper bounds

36. UNC at CHAPEL HILL & Ewha Womans University 36 Implementation and Results Timing measured on 2.8 GHz P4, 2G main memory Lower bound on PD g ♦ Pairwise PD t Upper bound on PD g ♦ UB1=Containment Optimization ♦ UB2=PD t (CH(A), CH(B))

37. UNC at CHAPEL HILL & Ewha Womans University 37 Performance Tris#28/1,692 8,452/336304/304 Convex Pieces 28/215 94/2844/44 Convex Separator 115343 t: LB (ms) 1.9014.3006.1274.112 t: UB 1 (ms) 21.664108.0241027.014482.511

38. UNC at CHAPEL HILL & Ewha Womans University 38 Hammer in Notch Hammer ♦ 1692 triangles ♦ 215 convex pieces Notch ♦ 28 triangles ♦ 28 convex pieces ♦ 1 convex separator Timings ♦ LB: 4.3 msec ♦ UB: 108 msec

39. UNC at CHAPEL HILL & Ewha Womans University 39 Comparison

40. UNC at CHAPEL HILL & Ewha Womans University 40 Hammer in Narrow Notch

41. UNC at CHAPEL HILL & Ewha Womans University 41 Comparison

42. UNC at CHAPEL HILL & Ewha Womans University 42 Spoon in Cup Cup ♦ 8452 triangles ♦ 94 convex pieces ♦ 53 convex separators Spoon ♦ 336 triangles ♦ 28 convex pieces

43. UNC at CHAPEL HILL & Ewha Womans University 43 Comparison

44. UNC at CHAPEL HILL & Ewha Womans University 44 Outline Previous Work PD g definition PD g algorithms ♦ Convex/Convex ♦ Convex/Convex complement ♦ Non-convex/Non-convex Application to motion planning

45. UNC at CHAPEL HILL & Ewha Womans University 45 Application to Motion planning C-obstacle query ♦ L. Zhang, Y.J. Kim, G.Varadhan, D. Manocha, Fast C-obstacle Query Computation for Motion Planning, ICRA 2006 C-space When the robot’s configuration changes within the C-space cell, Obstacle Workspace Whether the robot ‘escape’ from the obstacle at some moment? cell

46. UNC at CHAPEL HILL & Ewha Womans University 46 PD g for C-obstacle Query Query criterion: ♦ If the motion of the robot is less than the overlap extent (PD g ), the robot can not escape from the obstacle Motion bound PD g Is Motion bound < PD g ?

47. UNC at CHAPEL HILL & Ewha Womans University 47 Time: 110s Results: Motion Planning

48. UNC at CHAPEL HILL & Ewha Womans University 48 Summary Definition of generalized PD Proved PD g = PD t for convex objects Pose PD g problem for convex/convex complement as convex containment optimization Algorithms for lower and upper bounds on PD g PD g for motion planning

49. UNC at CHAPEL HILL & Ewha Womans University 49 Limitations Computes a lower bound and a upper bound for non-convex polyhedra. Can not guarantee a global minimum. Convex decomposition and convex separator enumeration impacts on the performance.

50. UNC at CHAPEL HILL & Ewha Womans University 50 Ongoing and Future work A Simple Path Non-Existence Algorithm for low DOF robots ♦ [L. Zhang, Y.J. Kim, D. Manocha] WAFR2006 Theoretical side: ♦ Formulate D g metric in a computational tractable way for exact PD g computation. Practical side: ♦ Apply PD g for higher DOFs motion planning, dynamic simulation, and tolerance verification.

51. UNC at CHAPEL HILL & Ewha Womans University 51 Acknowledgements Army Research Office, DARPA/REDCOM NSF ONR Intel Corporation KRF, STAR program of MOST, Ewha SMBA consortium, the ITRC program (Korea)

52. UNC at CHAPEL HILL & Ewha Womans University Thank you! http://gamma.cs.unc.edu/pdg http://graphics.ewha.ac.kr

53. UNC at CHAPEL HILL & Ewha Womans University 53 Discussion: why difficult for PD g ? 3D polyhedra PD t PD g Minkowski sumConfiguration space Dimension 3D6D DOFs 3T3T + 3R Arrangement Computation PlanesNon-linear hypersurfaces Non-convexO(n 6 )O(n 12 ) Distance metricEuclidean distance Easy to compute D g distance Difficult to compute

54. UNC at CHAPEL HILL & Ewha Womans University 54 PD g (A,B)≠ PD g (B,A) A