Gripping Sheet Metal Parts at Vertices

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

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

12. Analysis of Free Motion • Any motion can be broken down to component motions: • Translations: ex, ey, ez • Rotations: rx, ry, rz • Any infinitesimal motion = sequence of components.

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

14. I III II IV 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: 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. z y x Form-Closure in the x-y plane • Test of 2D v-grip is applied. • Ensures that distance decreases for ex, ey, rz. • Next, we consider ex, ez, ry.

16. z y x 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 ex, ez, ry. • For infinitesimal motions, vertices still lie strictly in corresponding region.

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. • |dq/dl| = |tan(f)/l|

22. Suggested Metric • Sensitivity of orientation to relaxing of jaws. • Consider all values of |tan(f)/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. I III II IV Friction • Is it possible to extend by just adding friction cone to the regions?

25. Diagonal Planes at Contacts • Horizontality is used only when analyzing rx. • 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. h l

28. Benchmarking Part 1

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

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

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.