1. Gripping Sheet Metal Parts at Vertices K. Gopalakrishnan A Project for CS 287

2. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

3. Introduction Grooves in cylindrical jaws used to grip sheet- metal parts Jaw Part

4. Motivation Simple reliable grips. Form-Closure achieved in 3D. Self-aligning grips. Very small Footprint.

5. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

6. Related Work V-grips (Form-closure only in 2D). [Gopalakrishnan, Goldberg, 2002] Multi DOF grippers in Robotic Fixtureless Assembly.[Plut, Bone, 1997]

7. Related Work Form-Closure & Force Closure –[Mason, 2001] –[Rimon, Burdick, 1995 & 1996] Necessary & Sufficient Conditions (number of contacts) –[Realeaux, 1963] –[Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987] –[Markenscoff, 1990]

8. Related Work Caging Grasps [Rimon, Blake, 1999] Efficient Computation of Nguyen regions [Van der Stappen, Wentink, Overmars, 1999] Multi-DOF Grips for Robotic Fixtureless Assembly [Plut, Bone, 1996 & 1997]

9. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

10. Problem Definition We first analyze piecewise planar sheet metal parts. Assumptions: Frictionless contacts. Linear part edges (perimeter & holes). First order form-closure only. Jaws are vertical parallel cylinders with horizontal V-shaped grooves. Part thickness negligible.

11. V-shaped Grooves Consists of intersection of 2 frustums as shown: Forces exerted normal to each frustum at point of contact. Jaw Part

12. Analysis of Free Motion Any motion can be broken down to component motions: Translations: e x, e y, e z Rotations: r x, r y, r z Any infinitesimal motion = sequence of components.

13. Coordinate System x-z plane contains axes of jaws. Jaws close along x. Jaw axes parallel to z. x zy

14. Results from V-grips Step1:We consider a pair of concave vertices. Step2:At these vertices, we draw normals to the edges through the jaw’s center. Step3:We label the 4 regions as shown: I II IV III Theorem: Both jaws lie strictly in the other’s Region I means it is an expanding v-grip or Both jaws lie strictly in the other’s Region IV means it is a contracting v-grip.

15. Form-Closure in the x-y plane Test of 2D v-grip is applied. Ensures that distance decreases for e x, e y, r z. Next, we consider e x, e z, r y. x zy

16. Form-Closure in the x-z plane Test of 2D v-grip is applied in x-z plane. However, consider the jaws as the part and part as contacts. Ensures that distance decreases for e x, e z, r y. For infinitesimal motions, vertices still lie strictly in corresponding region. x zy

17. Rotation about x axis Rotation about x axis always increases. Reason: Only parts which are horizontal at contacts are considered.

18. Theorem Any sheet-metal part which is horizontal at the points of contact is held in Form-Closure if –Part is in expanding/contracting 2D v-grip for horizontal projection. –Jaws are held in contracting/expanding 2D v-grip by the part in the plane containing axes of jaws. These conditions are sufficient but not necessary.

19. Algorithm For every face of part, generate all pairs of concave vertices on faces parallel to it. Test each pair for Form-Closure. If in Form-Closure, add to list of grips. Sort list by quality metric.

20. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

21. Quality Metric for v-grips Based on sensitivity to relaxing of jaws. Maximum change in orientation with one jaw still at a vertex. |d  /dl| = |tan(  )/l|

22. Suggested Metric Sensitivity of orientation to relaxing of jaws. Consider all values of |tan(  )/l| for v-grips in x-y and x-z planes. Take maximum value for worst change in orientation. Intuitive but not rigorous.

23. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

24. Friction Is it possible to extend by just adding friction cone to the regions? I II IV III

25. Diagonal Planes at Contacts Horizontality is used only when analyzing r x. If distance can be shown to reduce otherwise, horizontal assumption can be removed. E.g. Plane contains a line parallel to y-axis.

26. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

27. Strength of Conditions Compare results generated by sufficient conditions with those generated by brute force. l h

28. Benchmarking Part 1

29. Comparison Minimum half-v angle of frustum l Sufficient Conditions Brute Force

30. Comparison Minimum half-v angle of frustum h Brute Force Sufficient Conditions

31. Outline Introduction and Motivation Related Work Gripping Sheet Metal Parts Quality Metric Extensions Analysis of Results Conclusions & Future Work

32. Future Work Necessary & Sufficient Conditions. Acquisition. Trajectory Prediction. Friction. Second-order form-closure. Design of jaws for part.