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# Capturing a Convex Object with Three Discs

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• 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 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 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 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 Example: Initial conﬁ guration b a P Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 6. 6 Example: Escape through b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 7. 7 Example: Immobilized b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 8. 8 Example: Turning Counter-Clockwise b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 9. 9 Example: Escape through b a P c Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 A View of Conﬁ guration Space B* A* P* +θ , , Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 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 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 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 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 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 33. 33 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 34. 34 Approximate Capture Region: , , a b Capturing a Convex Object with Three Discs Shripad Thite thite@uiuc.edu
• 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 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 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. Appendix A: Exact Algorithm
• 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. 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. 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. Appendix B: First-Order Conﬁguration Space
• 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. 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. 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. 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. 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. 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. 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. 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. 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. The End