1. 1 D-Space and Deform Closure: A Framework for Holding Deformable Parts K. “Gopal” Gopalakrishnan, Ken Goldberg IEOR and EECS, U.C. Berkeley.

2. 2 Workholding: Rigid parts Summaries of results –[Mason, 2001] –[Bicchi, Kumar, 2000] Form and Force Closure –[Rimon, Burdick, 1995] Number of contacts –[Reuleaux, 1963], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], [Markenscoff, 1990] Caging Grasps –[Rimon, Blake, 1999] [Mason, 2001]

3. 3 Workholding: Rigid parts Nguyen regions –[Nguyen, 1988] Immobilizing three finger grasps –[Ponce, Burdick, Rimon, 1995] C-Spaces for closed chains –[Milgram, Trinkle, 2002] Fixturing hinged parts –[Cheong, Goldberg, Overmars, van der Stappen, 2002] Contact force prediction –[Wang, Pelinescu, 2003] [Mason, 2001] +- +- + + - -

4. 4 C-Space C-Space (Configuration Space): [Lozano-P’erez, 1983] Dual representation of part position and orientation. Each degree of part freedom is one C-space dimension. y x  /3 (5,4) y x q 4 5  /3 (5,4,-  /3)

5. 5 Avoiding collisions: C-obstacles Obstacles prevent parts from moving freely. Images in C-space are called C-obstacles. Rest is C free. Part Obstacle Part Obstacle

6. 6 Workholding and C-space Multiple contacts. 1 Contact = 1 C-obstacle. C free = Collision with no obstacle. Surface of C-obstacle: Contact, not collision.

7. 7 Form Closure A part is grasped in Form Closure if any infinitesimal motion results in collision. Form Closure = an isolated point in C-free. Force Closure = ability to resist any wrench. Part

8. 8 Bounded force-closure - [Wakamatsu, Hirai, Iwata, 1996] Manipulation of thin sheets - [Kavraki et al, 1998.] Robust manipulation - [Wada, Hirai, Mori, Kawamura, 2001] Holding Deformable Parts [Wada et al]

9. 9 Deformable parts “Form closure” does not apply: Can always avoid collisions by deforming the part.

10. 10 Deformation Space: A Generalization of Configuration Space. Based on Finite Element Mesh. D-Space

11. 11 Mesh M Part E Deformable Polygonal parts: Mesh Planar Part represented as Planar Mesh. Mesh = nodes + edges + Triangular elements. N nodes Polygonal boundary.

12. 12 D-Space A Deformation: Position of each mesh node. D-space: Space of all mesh deformations. Each node has 2 DOF. D-Space: 2N-dimensional Euclidean Space.

13. 13 D-Space: Example Simple example: 4-noded mesh. D-Space: 8-dimensional Euclidean Space. 2D slices show each mesh node’s position. Node positions also indicate part orientation. 13 42

14. 14 D-Obstacles No collision Collision 13 42 Slice of complement of D-obstacle (DA i ). Nodes 1,2,3 fixed.

15. 15 Topology violating deformation Undeformed part Allowed deformation Self collisions and D Topological

16. 16 Free Space: D free Slice with nodes 1-4 fixed Part and mesh 1 23 5 4 x y Slice with nodes 1,2,4,5 fixed x 3 y 3 x 5 y 5 x 5 y 5 x 5 y 5

17. 17 Modeling Forces Nodal displacement X: Vector of nodes’ displacement in global frame. Distance preserving transformation. X = T (q - q 0 ) Stiffness K: F = KX. Linear Elasticity. Nodal displacement X: Vector of nodes’ displacement in global frame. Distance preserving transformation. X = T (q - q 0 )

18. 18 Nominal mesh configuration Deformed mesh configuration Potential Energy Nodal displacement: Distance preserving transformation. X = T (q - q 0 ) q0q0 q For FEM with linear elasticity and linear interpolation, U(q q 0 ) = (1/2) X T K X

19. 19 Equilibrium Deformations Equilibrium: Local minimum of U. Stable equilibrium Strict local minimum of U. qAqA qBqB q U(q)

20. 20 Releasing the Part. Part should slide back to original deformation. Minimum work of U A needs to be done to release part. Caging grasps, saddle points [Rimon99] qAqA qBqB q U(q) UAUA

21. 21 Deform Closure Stable equilibrium = Deform Closure where U A > 0. qAqA qBqB q U(q)

22. 22 Independence from global coordinate frame. Proved by showing invariance of: - Deformation. - Potential energy and work. - Continuity in D-space. Theorem: Frame Invariance M E x1x1 y1y1 x2x2 y2y2

23. 23 Form-closure of rigid part Theorem: Equivalence  Deform-closure of equivalent deformable part. 

24. 24 Numerical Example 4 Joules547 Joules

25. 25 D-Obstacle symmetry Obstacle identical for all mesh triangles. Prismatic extrusions. Symmetry in D-Space 1 32 4 5 1 3 2 4 5

26. 26 Topology preservation symmetry. Define D' T - No mesh collisions. - No degenerate triangles. D T  D' T. Mirror images: - No continuous path. D' T identical for pairs of mesh triangles. Symmetry in D-Space 1 32 4 5 4 23 1 5

27. 27 Optimal 2-finger deform closure: Given jaw positions. Determine optimal jaw separation  *. Future work 

28. 28 If Quality metric Q = U A : Quality Metric

29. 29 Quality metric Plastic deformation:

30. 30 Q = min { U A, U L } Stress Strain Plastic Deformation LL Quality metric

31. 31 Holding multiple parts: Fixturing sheet metal parts for welding. Relative displacements of nodes. Quadratic programming approach. Future work

32. 32 http://alpha.ieor.berkeley.edu