1 Computing Deform Closure Grasps K. “Gopal” Gopalakrishnan, Ken Goldberg IEOR and EECS, UC Berkeley

2 Related Work D-Space and Deform Closure Two-point deform closure grasps Algorithm Outline

3 Workholding: Rigid parts Contact Mechanics: Number of contacts –[Reuleaux, 1876], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], –[Nguyen, 1988] –[Markenscoff, Papadimitriou, 1990] –[Han, Trinkle, Li, 1999] Immobility, 2 nd Order Form Closure –[Rimon, Burdick, 1995, 1998] –[Ponce, Burdick, Rimon, 1995] [Mason, 2001]

4 Workholding: Rigid parts Caging Grasps –[Rimon, Blake, 1999] Summaries of results –[Bicchi, Kumar, 2000] –[Mason, 2001] C-Spaces for closed chains –[Milgram, Trinkle, 2002] Fixturing hinged parts –[van der Stappen et al, 2002] Antipodal Points for Curved Parts –[Jia 2002]

5 C-Space C-Space (Configuration Space): [Lozano-Perez, 1983] Dual representation Each degree of part freedom is one C-space dimension. y x  /3 (5,4) y x  (5,4,-  /3) Physical space C-Space

6 Elastic Fingers, Soft Contacts [Hanafusa and Asada, 1982] [Salisbury and Mason, 1985] [Kim, Hirai, Inoue, 2003] Physical Models [Joukhadar, Bard, Laugier, 1994] Bounded Force Closure [Wakamatsu, Hirai, Iwata, 1996] Learned Grasps of Deformable Objects [Howard, Bekey, 1999] Deformable Parts

7 Path Planning for flexible wires, Elastic Sheets and Bodies [Kosuge et al, 1995] [Kavraki et al, 1998, 2000] [Amato et al, 2001] [Moll and Kavraki, 2004] Robot manipulation [Henrich and Worn, 2000] Robust manipulation with Vision [Hirai, Tsuboi, Wada 2001]

8 Modified spring-mass modeling [Yasuda, Yokoi, 2001] Holding sheet metal parts with dexterous parameter estimation [Ceglarek et al, 2002] Haptic rendering of deformations [Laycock, Day, 2003] Simulation of human hand deformation [Latombe et al, 2003] Deformable Parts

9 Related Work D-Space and Deform Closure Two-point deform closure grasps Algorithm Outline

10 Inspiration: Earlier work Grasping at concavities: –2D v-grips –3D v-grips

11 Inspiration: Earlier work Unilateral fixtures –Sheet metal parts.

12 Deformable parts “Form closure” insufficient: The part can always escape if deformed.

13 Inspiration: Earlier work Surgery simulation.

14 Lack of definition of fixtures/grasps for deformable parts. Generalization of C-Space. Inspiration

15 C-Space: Avoiding Collisions Finger body A in physical space is mapped on to the C- obstacle, CA. In C-space, the part shrinks to a point and the obstacle grows accordlingly CA c = C free. Physical space C-Space x y

16 Form Closure A part is grasped in Form Closure if any infinitesimal motion results in collision. Form Closure is equivalent to an isolated point in C-free. Physical space C-Space x y

17 Part Model Planar Polygonal Boundary Triangular Mesh Nodes (like hinges) Edges (like struts) Elements (deformable)

18 Deformation Space (D-Space) Each node has 2 DOF Analogous to configurations in C-Space D-Space: 2N-dimensional space of mesh node positions. point q in D-Space is a “deformation” q 0 is initial (undeformed) point 30-dimensional D-space

19 D-Space: Example Example part: 3-node mesh, 2 fixed. D-Space: 2D x y Physical space D-Space q0q0

20 Topology violating deformation Undeformed part Allowed deformation Self Collisions

21 D T : Topology Preserving Subspace x y Physical space D-Space D T  D-Space. DTDT DTC:DTC:

22 D-Obstacles x y Physical space D-Space Collision of any mesh element with obstacle. A Physical obstacle A i defines a deformation- obstacle DA i in D-Space. A1A1 DA 1

23 D-Space: Example Physical space x y D-Space Like C free, we define D free. D free = D T  [  (DA i C )]

24 With more nodes: D free Slice with only node 5 moving. Part and mesh x y Slice with only node 3 moving. x 3 y 3 x 5 y 5 x 5 y 5 x 5 y 5 Physical space D-Space D free = D T  [  (DA i C )]

25 Displacement between a pair of deformations q 0 : part’s nominal shape. X = q - q 0 : vector of nodal translations. Equivalent to moving origin in D-Space to q 0 and aligning nodal reference frames. q0q0 q X

26 Potential Energy Assume Linear Elasticity, Zero Friction K = FEM stiffness matrix. (2N  2N matrix for N nodes.) Forces at nodes: F = K X. Potential Energy: U(q) = (1/2) X T K X

27 Potential Energy “Surface” U : D free  R  0 Equilibrium: q where U is at a local minimum. In absence of friction or inertia, part will come to rest at an equilibrium. Stable Equilibrium: q where U is at a strict local minimum. = “Deform Closure Grasp” q U(q)

28 Potential Energy Needed to Escape from a Stable Equilibrium Consider: Stable equilibrium q A, Equilibrium q B. “Capture Region”: K(q A )  D free, such that any configuration in K(q A ) returns to q A. Saddlepoints [Rimon, Blake, 1995] q A qBqB q U(q) K( q A )

29 U A (q A ) = Increase in Potential Energy needed to escape from q A. = minimum external work needed to escape from q A. U A : Quality Measure q A qBqB q U(q) UAUA Potential Energy Needed to Escape from a Stable Equilibrium K( q A )

30 U A : Example U A = 4 JoulesU A = 547 Joules

31 Deform-closure: Frame invariant. Equivalence of form-closure and deform-closure. Symmetries in D T and DA i to simplify computation. D-Space: Analysis and Results x y x y D1:D1: D2:D2:

32 Related Work D-Space and Deform Closure Two-point deform closure grasps Algorithm Outline

33 Given: Pair of contact nodes. Determine: Optimal jaw separation. Optimal? Problem Description M E n0n0 n1n1 

34 Naïve Quality metric If Quality metric Q = U A. Maximum U A trivially at   = 0

35 Quality metric Plastic deformation. Occurs when strain exceeds e L.

36 Plastic Deformation Stress Strain Plastic Deformation eLeL A B C

37 New Quality Metric Additional work U L done by jaws for plastic deformation. New Q = min { U A, U L }; maximize Q Stress Strain Plastic Deformation eLeL A B C D Grasp U L = U (D) – U (Grasp)

38 Additional input: e L : Elastic limit strain.  : allowed error in quality metric. Additional assumptions: Sufficiently dense mesh !! Linear Elasticity. No Friction. Problem Description M, K E n0n0 n1n1 

39 No collisions of mesh. Monotonicity of U between mesh nodes. q1q1 q2q2 q1q1 q2q2 U Assumptions

40 Related Work D-Space and Deform Closure Two-point deform closure grasps Algorithm Outline

41 Paths and Points of Interest in D-Space

42 q1q1 q2q2 U ( q 1 )  U ( q 2 ) Paths and Points of Interest in D-Space

43 Paths and Points of Interest in D-Space

44 Potential Energy vs.  nini njnj k ij Potential Energy (U) Distance between FEM nodes Undeformed distance Expanding Contracting

45 Points of interest: contact at mesh nodes. Construct a graph: Each graph vertex = 1 pair of perimeter mesh nodes. p perimeter mesh nodes. O(p 2 ) graph vertices. Contact Graph

46 A B C E F G D Contact Graph: Edges Adjacent mesh nodes: A B C D E F G H H

47 Contact Graph Potential Energy Jaw 1 position Jaw 2 position

48 Contact Graph: Edges Non-adjacent mesh nodes: 

49 Contact Graph Potential Energy Jaw 1 position Jaw 2 position Sparse graph.

50 Traversal with minimum increase in energy. FEM solution with two mesh nodes fixed. nini njnj Deformation at Points of Interest

51  U ( v(n i, n j ),  ) Peak Potential Energy Given release path

52 Peak Potential Energy: All release paths  U ( v *,  )

53  U ( v o,  ), U ( v*,  ) Threshold Potential Energy U ( v*,  ) U ( v o,  ) U A (  ) U A (  ) = U ( v*,  ) - U ( v o,  )

54  U A (  ), U L (  )  Quality Metric U A (  ) U L (  ) Q (  )

55 Possibly exponential number of pieces. Sample in intervals of . Error bound on max. Q =  * max {  0 (n i, n j ) * k ij } Numerical Sampling  Q (  )  

56 Calculate U L. To determine U A : Algorithm inspired by Dijkstra’s algorithm for sparse graphs. Fixed  i

57 Algorithm for U A (  i )

58 Algorithm for U A (  i )

59 U Vertex v (traversed on path of minimum work) U(v) U(v*) Estimates for maximum U

60 Numerical Example Foam rubber part. Mesh and FEM using I-DEAS and ANSYS.

61 Numerical Example High elastic limit.

62 Numerical Example Undeformed  = 10 mm. Optimal   = 5.6 mm.

63 Future work Non-linear FEM. Mesh density: - Minimal meshes. - Special class of shapes? - Equivalent deformations between meshes?

64 Future work Mesh density: - Equivalent deformations between meshes? - Mapping D-spaces between meshes.

65 Future work Mesh density: - Error bounds. A A B B

66 Future work Optimal node selection.

67 Future work Friction