1. 1 On the Existence of Form- Closure Configurations on a Grid A.Frank van der Stappen Presented by K. Gopalakrishnan

2. 2 Inspiration Introduction Form Closure on a Grid Extensions Conclusion Outline

3. 3 Modular fixturing Existence of Fixtures –[Zhuang, Goldberg, 96] Purely geometric approach Inspiration

4. 4 Introduction & Related Work Form Closure on a Grid Extensions Conclusion Outline

5. 5 Ways to hold parts Form Closure –Any part motion causes collision Force Closure –Any external Wrench resisted by applying suitable forces [Mason, 2001]

6. 6 C-Space C-Space (Configuration Space) Describes position and orientation. We represent each degree of freedom of a part as a C-space coordinate. y x  /3 (5,4) y x  4 5  /3 (5,4,-  /3)

7. 7 Obstacles Obstacles prevent parts from moving freely. These images of obstacles in C- space are called C-obstacles. The rest is Free Space.

8. 8 Form Closure Form Closure: All adjacent points are collisions Difficult to Determine. Hence, distance from C-obstacles is used. [Rimon & Burdick, 96]

9. 9 Form Closure in C-space “Active” obstacles only Small motion cannot result in reduction of distance. Distance not easy to compute either.

10. 10 First order Form-Closure Consider Infinitesimal motion. Taylor Expansion for distance. Truncate to First order. Equivalent to replacing surfaces by tangents.

11. 11 First order Form-Closure In n dimensions there are n(n+1)/2 DOF. n translations n(n-1)/2 rotations For first order form-closure, n(n+1)/2+1 are necessary and sufficient –[Realeaux, 1963] –[Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987] –[Markenscoff, 1990]

12. 12 Fast Test for First Order Form-Closure Any infinitesimal motion on the plane is a rotation. No center of rotation possible for a part in Form-Closure. Try to identify possible centers.

13. 13 Fast Test +-

14. 14 Fast Test +- -+

15. 15 Fast Test +- -+

16. 16 Fast Test +- -+ + + - -

17. 17 Failure of First order Form- Closure Higher order terms neglected. Uncertainty when first order term is 0. We need to look at Second order terms. For generic parts in 2 or 3 dimensions, 3 or 4 point contacts are sufficient. [Rimon, Burdick, 1995]

18. 18 Second Order Form-Closure Example:1 st order approximation:

19. 19 Second order Form-Closure First order approximation allows a pure rotation about the centroid.

20. 20 Inspiration Introduction Form Closure on a Grid Extensions Conclusion Outline

21. 21 Problem Definition Rigid polygonal part. Frictionless point contacts. No parallel edges. To prove that: The part can be held in form-closure by 4 point-contacts that lie on 2 perpendicular lines.

22. 22 Notation PPolygonal Part eany edge of P aany point on e l e (a)normal to e at a

23. 23 Proof Consider the largest inscribed circle C in P. C has center m Let this touch the part at a 1, a 2, a 3. a1a1 a2a2 a3a3 m

24. 24 Proof 3 radial vectors positively span R 2. m only possible center of rotation. Contacts obtained in neighborhood of a i. a1a1 a2a2 a3a3 m

25. 25 Case I At least 1 a i is a vertex. Has to be Concave. a1a1 a2a2 a3a3 m

26. 26 Case I If no normal at vertex passes thru m, we are done a1a1 a2a2 a3a3 m

27. 27 Case I If a normal at the vertex passes through m, a1a1 a2a2 a3a3 m

28. 28 Case I

29. 29 Case II All 3 contacts at edges. Again, m is the only possible Let the least angle between radii be between ma 1 and ma 3. We choose a 4 to make a 1 a 4 and a 2 a 3 gridlines.

30. 30 Case II-1 m is to the left of normal at a 4. Move the horizontal line slightly upward.

31. 31 Case II-2 m is to the right of normal at a 4. Rotate both axes by equal angles.

32. 32 Case II-3 m is to on the normal at a 4. Coordinated translation of horizontal and vertical gridlines. + - + - + - + -

33. 33 Inspiration Introduction Form Closure on a Grid Extensions Conclusion Outline

34. 34 Parallel Edges Counterexamples. Topology not maintained.

35. 35 Contact at Grid Vertex Not feasible (e.g. convex polygons). Additional contact on another line required. Feasibility not guaranteed. Requires large enough part.

36. 36 Jaws with non-zero radii Jaw has a radius r The part is transformed with a Minkowsky addition, offsetting the polygons with a disk of radius r.

37. 37 Jaws with non-zero radii Contact at Convex vertex not desired. As radius increases, Minkowsky sum becomes disc.

38. 38 Inspiration Introduction Form Closure on a Grid Extensions Conclusion Outline

39. 39 Conclusion Existence of solution proved under assumptions. Validity of assumptions shown. Non zero radius incorporated. Extension to non-polygonal parts open. Accessibility to clamps ignored.

40. 40