CAGING OF RIGID POLYTOPES VIA DISPERSION CONTROL OF POINT FINGERS

1. CAGING OF RIGID POLYTOPES VIA DISPERSION CONTROL OF POINT FINGERS PeamPipattanasompornAdvisor: AttawithSudsang

2. Motivation? ! ! !

3. Better Approach?

4. Overview Proposed Ph.d. Thesis Additional Chapters Master Thesis 2-Squeeze(2006) n-Squeeze(2008) L X H Fix Cage(2011) S C Robust Cage(2012) 2-Stretch(2006) n-Stretch(2008) Imperfect Shape (2010)

5. Overview Proposed Ph.d. Thesis Additional Chapters Master Thesis 2-Squeeze(2006) n-Squeeze(2008) L X H Fix Cage(2011) S C Robust Cage(2012) 2-Stretch(2006) n-Stretch(2008) Imperfect Shape (2010)

6. 2-Squeeze, How? • Keep distance below a value • Given object shape, solve: • Where to place the fingers? • The upperbounddistance? H “Distance”

7. 2-Squeeze • Possible escape path (object frame) H Distance Along the path

8. 2-Squeeze • “Better” escape path H Distance Upperbound “Better” Initial Along the path

9. 2-Squeeze • Find an Optimal Escape Path in C-Free H (abstracted)C-Obstacle Abstracted set ofescape configurations a (a,b) b Configuration Space (4D) Workspace (2D)

10. 2-Squeeze • Find an Optimal Escape Path in C-Free (abstract)C-Obstacle Abstract set ofescape configurations (a,b) Configuration Space (4D)

11. C-Free Decomposition C-Obstacle

12. Paths connecting Terminals C-Obstacle

13. Finite Categorization of Paths C-Obstacle

14. Straight Path Distance : |a-b|2 a b (linear interpolation) Along the path (a,b)

15. Moving Across Convex Subsets C-Obstacle

16. Through Convex Intersections C-Obstacle

17. Requirements For The Algorithm Distance(x) & x Convex Rigid Transformation Invariant

18. Convex & RTI Examples • d1+ d2+ d3 • d12+ d22+ d32 • max(d1, d2, d3) x1 d3 d1 x3 x2 d2 • Larger  Loose cage • Fingers at a point  Smallest “Formation Size”

19. Results (n-Squeeze) Size: d12+d22+d32+d42

20. Squeezing? 1 1 1 1 3 3 2 2 3 2 3 2 1-DOF Scaling ONLY

21. “Size” & “Deformation” Smaller sizeSlightly Deformed 1 1 1 1 3 2 3 2 3 2 2 3 Reference Formation Same sizeNo deformation Larger sizeDeformed

22. “Size” & “Deformation” Smaller sizeSlightly Deformed 1 1 1 1 3 2 3 3 2 2 2 3 Reference Formation Same sizeNo deformationSame Formation Larger sizeDeformed

23. “Size” & “Deformation” Smaller sizeSlightly Deformed 1 1 1 1 3 2 3 3 2 2 2 3 Reference Formation Same sizeNo deformationSame Formation Larger sizeDeformed

24. “Size” & “Deformation” Smaller sizeSlightly Deformed 1 1 1 1 3 2 3 3 2 2 2 3 Reference Formation Same sizeNo deformationSame Formation Larger sizeDeformed

25. “Size” & “Deformation” • |r|22(x) = |A†x|22 • “Scale” or “Size” (w.r.t. reference) • D(x) = |A(r;t) – x|22 • “Deformation upto Scale” (w.r.t. reference) 1 1 3 2 2 3 A stores information of the reference.

26. Squeezing ? Convex& RTI Size = |r|22<??? D ≤ 0 & 1 1 1 3 2 3 2 3 2 Convex& RTI 1-DOF Scaling ONLY

27.  Squeezing |r|22; D ≤ 0  ; D > 0 Size* = Size = |r|22 <??? D ≤ 0  &  1 1 1 D ≤ 0 D > 0 D > 0 3 2 3 2 3 2 x 1-DOF Scaling ONLY

28. Fix Formation Cage Convex& RTI ConvexConstraint Size* = 1 1 Size* ≤ 1 Size*  1 3 2 & “Squeeze” “Stretch”

29. Robust Caging • Keep error (deformation) below a value • Given object shape, find: • Where to place the fingers • The upperbounderror Independent Capture Regions

30. n-Squeeze vs Fix Formation X KEEP SIZE ERROR (DEFORMATION) BELOW UPPERBOUND BELOW UPPERBOUNDOPTIMAL ESCAPE PATH SIZE MINIMIZE UPPERBOUND DISTANCE ERROR (DEFORMATION)

31. Error Tolerance  1 1 1 1 • r+t- inf D2 =  2 2 2 2 r,tϵR2  3 3 3 3 2 “Placement Error upto Scale” “Placement Error”  1 1 1 1 inf Ep = • r+t-  2 2 2 2 |r|2=1tϵR2  3 3 3 3 p NOT CONVEX!

32. Approximation  infg(r)r ϵRi infg(r)|r|2=1  mini ϵ{1,…, m} R2 R3 R1 R4

33. Approximation  infg(r)r ϵRi infg(r)|r|2=1  mini ϵ{1,…, m} R2 Min of Convex Functions(not convex) R3 R1 R4

34. Optimal Path Min of a Convex Function is Convexf = f1 = min(f1)

35. Optimal Path f1 = f f = f2 f1 = f = f2 Min of Two Convex Functionsf = min(f1, f2) 35

36. Optimal Path f1 = f f = f2 f1 = f = f2 ??? Min of Two Convex Functionsf = min(f1, f2) 36

37. Optimal Path What is the optimal path, starting from the minimal points? f1=f=f2 f(x) f=f1 f=f2 2 1 x

38. Critical Point Consider… f1=f=f2 f(x) f=f1 f=f2 1,2 2 1 x Only the points under the water level are reachable when the maximum deformation is limited to below the water level.

39. Optimal Path : minimizer for a CONVEX optimization problem:minimize L s.t.f1(x) < Lf2(x) < L f1=f=f2 f(x) f=f1 f=f2 1,2 Critical Value 2 1 x

40. Critical Point f1=f=f2 f(x) f=f1 f=f2 1,2 Critical Value 2 1 x

41. Min of Multiple Convex Functions f= f2 f= f1 f= f3 Min of Multiple Convex Functionsf = min(f1, f2 , f3)

42. Min of Multiple Convex Functions 2,3 f= f2 1,2 2 3 f= f1 f= f3 1,3 1 Min of Multiple Convex Functionsf = min(f1, f2 , f3)

43. Search Space 2 1,2 1,3 1 3 2,3 Min of Multiple Convex Functionsf = min(f1, f2 , f3) Include all possible between any two regions: f=fi, f=fj

44. Optimal Path

45. Results

46. Results

47. Shape Uncertainty Scanned Object Exact Object(Unknown) sensor

48. Idea • Cage subobject Cage object ? • Fingers must not penetrate the object. H

49. Idea • Find placements that cage subobject, outside superobject. Exact Object(Unknown) Exact boundary (unknown) but inbetween the bounds.

50. Applications • Simplification • Curved Surface, Spherical Fingers • Shape Uncertainty • Slightly Deformable Object • Partial Observation