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 Number of contacts –[Reuleaux, 1876], [Somoff, 1900] –[Mishra, Schwarz, Sharir, 1987], [Markenscoff, 1990] Nguyen regions –[Nguyen, 1988] Form and Force 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 –[Cheong, Goldberg, Overmars, van der Stappen, 2002] Contact force prediction –[Wang, Pelinescu, 2003]

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  4 5  /3 (5,4,-  /3) Physical space C-Space

5 Avoiding Collisions: C-obstacles Obstacles prevent parts from moving freely. Images in C-space are called C-obstacles. Rest is C free. Physical space C-Space x y

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. 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 = an isolated point in C-free. Force Closure = ability to resist any wrench. Physical space C-Space x  y

8 Grasp planning: Combining Geometric and Physical models - [Joukhadar, Bard, Laugier, 1994] Bounded force-closure -[Wakamatsu, Hirai, Iwata, 1996] Minimum Lifting Force - [Howard, Bekey, 1999] Holding Deformable Parts

9 Manipulation of flexible sheets -[Kavraki et al, 1998] Quasi-static path planning. - [Anshelevich et al, 2000] Robust manipulation - [Wada, Hirai, Mori, Kawamura, 2001]

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

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

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

13 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. 30-dimensional D-space

14 Nominal mesh configuration Deformed mesh configuration Deformations Deformations (mesh configurations) specified as list of translational DOFs of each mesh node. Mesh rotation also represented by node displacements. Nominal mesh configuration (undeformed mesh): q 0. General mesh configuration: q. q0q0 q

15 D-Space: Example Simple example: 3-noded mesh, 2 fixed. D-Space: 2-dimensional Euclidean Space. D-Space shows moving node’s position. x y Physical space D-Space q0q0

16 Topological Constraints: D T x y Physical space D-Space Mesh topology maintained. Non-degenerate triangles only. DTDT

17 Topology violating deformation Undeformed part Allowed deformation Self Collisions and D T

18 D-Obstacles x y Physical space D-Space Collision of any mesh triangle with an object. Physical obstacle A i has an image DA i in D-Space. A1A1 DA 1

19 D-Space: Example Physical space x y D-Space D free = D T  [  (DA i C )]

20 Free Space: 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

21 Nodal displacement X = q - q 0 : vector of nodal translations. Equivalent to moving origin in D-Space to q 0. D- space q0q0 q

22 Potential Energy 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

23 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. Stable Equilibrium = “Deform Closure Grasp” q U(q)

24 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 )

25 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 : Measure of “Deform Closure Grasp Quality” q A qBqB q U(q) UAUA Potential Energy Needed to Escape from a Stable Equilibrium K( q A )

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

27 Theorem: Definition of Deform closure grasp and UA is 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:

28 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

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

30 Numerical Example 4 Joules547 Joules

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

32 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

33 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

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

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

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

37 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

38 Potential Energy Surface Potential Energy Jaw 1 position Jaw 2 position

39 Algorithm for U A (  i )

40 Algorithm for U A (  i )

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

42 Numerical Example Undeformed  = 10 mm. Optimal   = 5.6 mm. Foam Rubber e L = 0.8. FEM performed using ANSYS. Computing Deform Closure Grasps, K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct

43 x y z 3D Meshes Tetrahedral elements: - 3 DOF per node. Box elements: - Translational + Rotational DOF. Sheet metal: - Shell elements.

44 Framework for describing all deformations. Defined Deform Closure. Proved Frame Invariance and equivalence to Form-Closure. Exploiting symmetry in computing D free. Optimal two-point deform closure grasps. Summary