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

2 Workholding: Rigid parts Contact Mechanics: Number of contacts –[Reuleaux, 1876], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], –[Nguyen, 1988] –[Markenscoff, Papadimitriou, 1990] Immobility, 2 nd Order Form Closure –[Rimon, Burdick, 1995] Immobilizing three finger grasps –[Ponce, Burdick, Rimon, 1995] [Mason, 2001]

3 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]

4 C-Space C-Space (Configuration Space): [Lozano-Perez, 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  (5,4,-  /3) Physical space C-Space

5 Avoiding Collisions: C-obstacles Consider: a blue part collides with a point Obstacle A at a set of configurations. The set in C-space is 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

6 Workholding and C-space “Finger bodies” are obstacles. For planar rigid part, C-space has 3 dimensions. Physical space C-Space x  y

7 Form Closure A part is grasped in Form Closure if any infinitesimal motion results in collision. Form Closure = isolated point in C-free. Physical space C-Space x  y

8 Elastic Fingers [Hanafusa and Asada, 1982] Physical Models [Joukhadar, Bard, Laugier, 1994] Bounded Force Closure [Wakamatsu, Hirai, Iwata, 1996] Learned Grasps of Deformable Objects [Howard, Bekey, 1999] Holding Deformable Parts

9 Path Planning for Elastic Sheets and Bodies [Kavraki et al, 1998, 2000] Fabric Handling [Henrich and Worn, 2000] Robust manipulation with Vision [Hirai, Tsuboi, Wada 2001]

10 Deformable parts “Form closure” insufficient: Can always escape by deforming the part.

11 Deformable Polygonal parts Modeling Deformation: Planar Mesh N Nodes Edges Triangular elements. Polygonal perimeter.

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

13 D-Space: Example Example part: 3-node mesh, 2 fixed. D-Space: 2D Nominal deformation: q0. Deformations in D-Space x y Physical space D-Space q0q0

14 Topology violating deformation Undeformed part Allowed deformation Self Collisions

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

16 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

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

18 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 )]

19 Displacement between a pair of deformations X = q - q 0 : vector of nodal translations. Equivalent to moving origin in D-Space to q 0. q0q0 q X

20 Potential Energy Assume Linear Elasticity. 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

21 Potential Energy “Surface” U : D free  R  0 Equilibrium: q where U is at a local minimum. Stable Equilibrium: q where U is at a strict local minimum. = “Deform Closure Grasp” q U(q)

22 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. q A qBqB q U(q) K( q A )

23 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 )

24 Quality Measure Example (U A = Energy needed to Escape) U A = 4 JoulesU A = 547 Joules

25 Theorem: Definitions of Deform closure and UA are frame-invariant. Proof: Consider D-spaces D 1 and D 2. - Consider q 1  D1, q 2  D 2. such that physical meshes are identical. Theorem 1: Frame Invariance x y x y D1:D1: D2:D2:

26 There exists distance preserving linear transformation T such that q 2 = T q 1. It can be shown that U A2 (q 2 ) = U A1 (q 1 ) Theorem 1: Frame Invariance x y x y

27 Form-closure of rigid part Theorem 2: Form Closure and Deform Closure  Deform-closure of equivalent deformable part. 

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

29 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 prism. The shape of DA e is same for all elements. Computing DA i

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

31 Two-point Deform Closure: Given contact nodes: Determine optimal jaw separation  *. Future work 

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

33 New Quality Metric Plastic deformation. Occurs when strain exceeds e L.

34 New Quality Metric Additional work U L done by jaws for plastic deformation. New Q = min { U A, U L } Stress Strain Plastic Deformation A B C eLeL A B C

35 Potential Energy Surface Potential Energy Jaw 1 position Jaw 2 position

36 Algorithm for U A (  i )

37 Algorithm for U A (  i )

38 U Vertex v (traversed on path of minimum work) U(v) U(v*) Algorithm for U A (  i )

39 Numerical Example Undeformed  = 10 mm. Optimal   = 5.6 mm. Foam Rubber e L = 0.8. FEM performed using ANSYS.

40 x y z Extension to 3D Tetrahedral elements: - 3 DOF per node.

41 D-Space: a “C-Space”-like framework for holding deformable parts. D free = D T  [  (DA i C )] Potential Energy and Deform Closure Thm 1: Frame Invariance Thm 2: Form-Closure Equivalence Symmetry in D free. Quality Metrics, Optimal Deform Closure Summary