1 Computing Deform Closure Grasps K. “Gopal” Gopalakrishnan Ken Goldberg UC Berkeley

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

3 Holding: 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 Holding: Rigid parts 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] Caging Grasps – [Rimon, Blake, 1999]

5 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] Robot manipulation [Henrich and Worn, 2000] Robust manipulation with Vision [Hirai, Tsuboi, Wada 2001] Holding Deformable Parts

6 Deformable Parts Path Planning for Elastic Wires, Sheets and Bodies [Kavraki et al, 1998, 2000] [Amato et al, 2001] [Moll and Kavraki, 2004]

7 Context Grasping at concavities: –2D v-grips –3D v-grips

8 Context Unilateral fixtures –Sheet metal parts, constrained deformation:

9 Context Surgery simulation.

10 Deformable parts “Form closure” based on immobility For deformable parts, how to define immobility? The part can always escape:

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

12 Introduce FEM Mesh Define “C-Space” based on node displacements Characterize potential energy Idea

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

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

15 D-Space Example: 2 fixed nodes 1 moveable node: 2 dimensional D-Space x y Physical space D-Space q0q0

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

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

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

19 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 )]

20 Displacement between a pair of deformations q 0 : part’s nominal shape. X = q - q 0 : vector of nodal translations. q0q0 q X

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

22 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)

23 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 )

24 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 )

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

26 Definition of Deform closure is frame-invariant. Proof: Consider D-spaces D 1 and D 2. - Consider q 1  D 1, q 2  D 2. such that physical meshes are identical. Thm 1: Frame Invariance x y x y D1:D1: D2:D2:

27 There exists distance preserving linear transformation T such that q 2 = T q 1. and U A2 (q 2 ) = U A1 (q 1 ) U A does not depend on frame. Frame Invariance x y x y

28 Form-closure of rigid part Thm 2: Form Closure and Deform Closure  Deform-closure of equivalent deformable part.  If in form-closure, strict local minimum: work needed to escape. If in deform-closure, and no deformation allowed: form closure.

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

30 Given a pair of point jaws Determine “optimal” jaw separation. Problem Description M E n0n0 n1n1 

31 Naïve Quality metric Consider Quality metric Q = U A. Maximum U A trivially at   = 0

32 New Quality Metric Consider work needed before plastic deformation occurs: U L (Quadratic function of jaw separation) Let Q = min { U A, U L } Find jaw separation that maximizes Q Separation where: Work needed to release part = Work needed to reach elastic limit Stress Strain Plastic Deformation eLeL A B C D G U L = U (D) – U (G)

33 Input: Mesh, K matrix, Initial jaw contacts at perimeter nodes e L : Elastic limit strain.  : Approx. error Output: Jaw separation that optimizes Q Assumptions: Sufficiently dense mesh Linear Elasticity until E Limit Problem Description M, K n0n0 n1n1 

34 no self-collisionsmoving between perimeter nodes: U is monotone q1q1 q2q2 q1q1 q2q2 U Assumptions

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

36 Computing U A (minimum potential energy needed to release part) as a function of jaw separation: Consider escape paths in D-Space that maintain contact, find saddle points

37 Escape Paths in D-Space: motions between pairs of perimeter nodes

38 Construct Contact Graph: Each graph vertex = one pair of perimeter mesh nodes. p perimeter mesh nodes. p 2 graph vertices. Problem input is a vertex Compute escape paths “Contact” Graph: all perimeter node pairs

39 At each Graph vertex, Potential Energy is a Quadratic function of  nini njnj k ij We can compute the spring constant by running one FEM iteration: reduce jaw separation by one unit, resulting force yields stiffness constant Compute k for all graph vertices

40 A B C E F G D Contact Graph edges correspond to adjacent transitions between perimeter nodes A B C D E F G H H

41 Find escape paths along vertices in the Contact Graph

42  U ( v(n i, n j ),  ) Each vertex along the escape path defines a quadratic function of jaw separation. Upper envelope of these functions gives potential energy function for the path Computing U A (  )

43 Lower envelope of all escape paths corresponds to min potential energy needed to release part as function of   U ( v *,  ) Computing U A (  )

44  U A (  ), U L (  )  Recall Energy to reach Plastic Limit is Quadratic increasing: find crossover point U A (  ) U L (  ) Q (  ) Computing  *

45 U A may have an exponential number of pieces, so sample at uniform intervals d choose d based on given  and slope/stiffness of all energy curves Numerical Sampling  Q (  )  

46 Compute stiffness for each graph vertex, solving the FEM for mesh with n nodes to : O(n 3 ) Compute stiffness for all graph vertices: O(p 2 n 3 ) Compute optimal jaw separation: O(p 2 ( n 3 + (1/  ) log p)) Complexity

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

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

49 Future work Initial node pair selection Mesh dependencies: - Minimal meshes. -Equivalent deformations between meshes

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

51

52 Future work Friction

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

54 Future work Optimal node selection.

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

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

59 High Dimensional. Computing D T and DA i. Exploit symmetry. Computing D free DA i D free DTCDTC

60 Consider obstacle A and one triangular element. Consider the slice D e of D, corresponding to the 6 DOF of this element. Along all other axes of D, D e is constant. Extruded cross-section is a 6D prism. The shape of DA e is same for all elements. Computing DA i

61 Thus, DA is the union of identical prisms with orthogonal axes. Center of DA is the deformation where the part has been shrunk to a point inside A. Similar approach for D T. Computing DA i 

62 parameterize jaw positions along part perimeter, separation between jaws defines an energy surface (Blake, Rimon) Potential Energy Jaw 1 position Jaw 2 position

63  U ( v o,  ), U ( v*,  ) Subtract potential energy due to deformation by given jaw contacts U ( v*,  ) U ( v o,  ) U A (  ) U A (  ) = U ( v*,  ) - U ( v o,  ) This yields U A, potential energy needed to escape

64 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

65 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

66 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

67 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