Capturing a Convex Object with Three Discs

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Capturing a Convex Object with Three Discs

  1. 1. Capturing a Convex Object with Three Discs Jeff Erickson Shripad Thite Fred Rothganger Jean Ponce jeffe, thite, rothgang, jponce @uiuc.edu Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801, USA http://www-cvr.ai.uiuc.edu September 17, 2003 @ ICRA (Taipei, Taiwan)
  2. 2. 2 Problem b a P Input: – an arbitrary convex object in the plane – three disc-shaped robots— , , and Given: – fixed positions of robots and – position of in contact with and in its initial orientation Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  3. 3. 3 Problem b a P c Question: Where should robot be placed to capture ? Capture region the set of all positions (if any) of that prevent from escaping Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  4. 4. 4 Problem can perform arbitrary rigid motions The interior of cannot intersect any robot Capture prevent from escaping to infinity Capture grasp (immobilize) Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  5. 5. 5 Example: Initial confi guration b a P Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  6. 6. 6 Example: Escape through b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  7. 7. 7 Example: Immobilized b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  8. 8. 8 Example: Turning Counter-Clockwise b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  9. 9. 9 Example: Escape through b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  10. 10. 10 No Escape a b P c can spin around without being able to escape Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  11. 11. 11 Motivation http://www-cvr.ai.uiuc.edu/ponce grp/demo/robots/ Robots may be mobile platforms, fingertips of a gripper, pins of a parts feeder. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  12. 12. 12 Motivation Sensorless manipulation: in-hand manipulation mobile robot motion planning regrasping [Sundsang, Phoka: ICRA ’03, Session WP9] Some advantages: contact need not be maintained during manipulation no reliance on any particular model of friction tolerance of uncertainty Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  13. 13. 13 Related Work Fixturing: [Gopal, Goldberg et al., ICRA 2002] (quality of grasp = 1 / amount of wiggle room) Manipulating algebraic parts in the plane [Rao, Goldberg—ITRA ’95] Graspless manipulation [Lynch—ICRA ’97; Lynch ’96] Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  14. 14. 14 Previous Work Maximum Independent Capture Discs (MICaDs) [Sudsang, Ponce—ICRA ’98 and IEEE Trans. Rob. Autom. 2002]: associate each robot with a fixed edge of the object (the object is a triangle); starting from an immobilizing configuration, compute the maximum distance that one robot can travel while keeping object captured compute three maximum discs representing triples of capturing configurations Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  15. 15. 15 Previous Work Capturing a concave polygon with two robots [Sudsang, Luewirawong—ICRA 2003] Caging planar bodies with one-parameter two-fingered systems [Rimon, Blake—IJRR 2000] Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  16. 16. 16 Some Assumptions b a P c Without loss of generality robot is at the origin robot is on the positive -axis , , and are labeled in clockwise order Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  17. 17. 17 Simplifi cations for this Talk b a P c is a polygon Robots have zero radius (points) All definitions still apply when and the robots are arbitrary convex objects. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  18. 18. 18 Canonical Motion of b a c turns (counter)clockwise keeping in contact with and . One of three things happens Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  19. 19. 19 Canonical Motion of b a b a c c (i) achieves an escape angle in which it escapes by pure translation Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  20. 20. 20 Canonical Motion of b a b a c c (ii) is blocked from turning further by (a triple contact) Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  21. 21. 21 Canonical Motion of a b P c (iii) turns all the way around without being able to escape Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  22. 22. 22 Main Lemma Lemma: If can escape by any rigid motion, then can escape by canonical motion. Contra-positive: If cannot escape by canonical motion, then cannot escape at all. If can escape through , then in a preprocessing step we compute the first escape angle such that in orientation can escape through . Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  23. 23. 23 Capture Region the set of all positions of that prevent from escaping We will concentrate on CCW-escape and computing henceforth. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  24. 24. 24 Confi guration Space is the position of and is the orientation of Without loss of generality, the initial orientation is . Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  25. 25. 25 A -Slice of Confi guration Space B* A* θ θ b a Pθ c , Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  26. 26. 26 Preventing Escape by Pure Translation B* A* θ θ b a Pθ c in orientation can escape by pure translation in one of three ways: Escape through : if int (equiv., int ) Escape through : if int Escape through : if int Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  27. 27. 27 A View of Confi guration Space B* A* P* +θ , , Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  28. 28. 28 Visibility Problem Consider a fixed position of robot Let be the ray with origin at pointing in the -direction (upwards). What does intersect first? Let be the value of where the first intersection of occurs. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  29. 29. 29 Visibility Problem If , then is not captured; otherwise, intersects one of the following: case : blocks in orientation from turning further, i.e., is captured! case : int and can escape through case : int and can escape through Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  30. 30. 30 Exact Algorithm is the projection of the lower envelope of , , and . is the projection of the upper envelope of , , and . We can compute the capture region in polynomial time. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  31. 31. 31 Approximate Algorithm Compute the capture region restricted to a grid (a discrete approximation with any desired accuracy). Compute polyhedral approximations of , , and . Render each with a -buffer using orientation as the depth. The grid points where is visible constitute the approximate capture region (and similarly for ). Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  32. 32. 32 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  33. 33. 33 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  34. 34. 34 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  35. 35. 35 Conclusion We characterized the capture region of a three robot that capture a convex object in the plane in conjunction with two fixed robots. Reduced to a visibility problem in configuration space . Computed the exact capture region in polynomial time. Used graphics hardware ( -buffer) to compute a discrete approximation quickly. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  36. 36. 36 Directions for Future Work Capturing non-convex objects or linkages of convex objects (canonical motion? two robots suffice?) [Sudsang, Luewirawong—ICRA 2003] Integrating the computation of the capture region into algorithms for grasping, manipulation, and motion planning Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  37. 37. 37 Thanks Steve LaValle, Svetlana Lazebnik, and others NSF grants IRI-9907009, CCR-0093348, and CCR-0219594 Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  38. 38. Appendix A: Exact Algorithm
  39. 39. Appendix A: Exact Algorithm 39 Exact Algorithm The boundary of is a piecewise-smooth collection of algebraic surface patches of constant degree with no self-intersections. The number of patches (number of critical orientations) is . The critical orientations can be computed in time by a rotating calipers algorithm. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  40. 40. Appendix A: Exact Algorithm 40 Exact Algorithm Project boundary curves and silhouette curves of each of the surface patches on to the plane. Compute the arrangement of these curves with cells. The capture region is the union of cells in this cell decomposition. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  41. 41. Appendix A: Exact Algorithm 41 Exact Algorithm Compute a cylindrical decomposition of the volume between the lower envelope of the patches above and the upper envelope of patches below . The intersection of each cylinder is the union of cells in the cell decomposition—all these cells are marked once if the object immediately above is either and marked again if the object immediately below is . The capture region is the union of twice-marked cells. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  42. 42. Appendix B: First-Order Configuration Space
  43. 43. Appendix B: First-Order Configuration Space 43 Confi guration Space is the position and is the orientation of Without loss of generality, the initial orientation is . Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  44. 44. Appendix B: First-Order Configuration Space 44 First-Order Obstacles Fix the position of . A B p C , , Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  45. 45. Appendix B: First-Order Configuration Space 45 Escape by Pure Translation in orientation can escape by pure translation in one of three ways: Escape through : if Escape through : if Escape through : if Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  46. 46. Appendix B: First-Order Configuration Space 46 Escape by Pure Translation b a b b a a b b a a c c c c c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  47. 47. Appendix B: First-Order Configuration Space 47 Valid Confi gurations , , Invalid region = int int int Space of valid configurations int and int and int Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  48. 48. Appendix B: First-Order Configuration Space 48 Structure of Valid Space Every rigid motion of that does not intersect any robot corresponds to a path in valid space. is captured if and only if the component of valid space containing the initial configuration is compact (closed and bounded). If can escape, then there exists a such that in orientation can escape by pure translation. Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  49. 49. Appendix B: First-Order Configuration Space 49 Escape Orientation Necessary and sufficient condition for escape: some pair of obstacles do not intersect, i.e., either 1. , or 2. , or 3. (Counterclockwise) Escape angle = , or , or (If CCW escape is not possible, then .) Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  50. 50. Appendix B: First-Order Configuration Space 50 Canonical Motion Revisited Canonical motion of is motion along . Triple contact: Initial configuration is valid, i.e., . Counterclockwise escape angle: , or , or Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  51. 51. Appendix B: First-Order Configuration Space 51 Relation between Two Confi guration Spaces A B B* A* θ θ b a p Pθ C c (equiv., ) Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
  52. 52. The End

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