K' Gopal Gopalakrishnan, Ken Goldberg

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D-Space and Deform Closure A Framework
for Holding Deformable Parts

• K. Gopal Gopalakrishnan, Ken Goldberg
• IEOR and EECS, U.C. Berkeley.

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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
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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

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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.

Physical space
C-Space
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Avoiding Collisions C-obstacles

• Obstacles prevent parts from moving freely.
• Images in C-space are called C-obstacles.
• Rest is Cfree.

Physical space
C-Space
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Workholding and C-space

• Multiple contacts.
• 1 Contact 1 C-obstacle.
• Cfree Collision with no obstacle.
• Surface of C-obstacle Contact, not collision.

Physical space
C-Space
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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
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Holding Deformable Parts

• Grasp planning Combining Geometric and Physical
  models
• - Joukhadar, Bard, Laugier, 1994
• Bounded force-closure
• Wakamatsu, Hirai, Iwata, 1996
• Minimum Lifting Force
• - Howard, Bekey, 1999

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Holding Deformable Parts

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

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Deformable parts

• Form closure does not apply
• Can always avoid collisions by deforming the part.

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D-Space

• Deformation Space A Generalization of
  Configuration Space.
• Based on Finite Element Mesh.

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Deformable Polygonal parts Mesh

• Planar Part represented as Planar Mesh.
• Mesh nodes edges Triangular elements.
• N nodes
• Polygonal boundary.

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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
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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)
  q0.
• General mesh configuration q.

q0
Nominal mesh configuration
q
Deformed mesh configuration
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D-Space Example

• Simple example
• 3-noded mesh, 2 fixed.
• D-Space 2-dimensional Euclidean Space.
• D-Space shows moving nodes position.

Physical space
D-Space
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Topological Constraints DT

• Mesh topology maintained.
• Non-degenerate triangles only.

Physical space
D-Space
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Self collisions and DT
Allowed deformation
Undeformed part
Topology violating deformation
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D-Obstacles
A1

• Collision of any mesh triangle with an object.
• Physical obstacle Ai has an image DAi in D-Space.

Physical space
DA1
D-Space
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D-Space Example

• Dfree DT ? ? (DAiC)

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Free Space Dfree
1
4
5
2
3
Part and mesh
y
y
y
y
5
5
5
3
x
x
x
x
3
5
5
5
Slice with nodes 1-4 fixed
Slice with nodes 1,2,4,5 fixed
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Nodal displacement

• Displacements of mesh nodes (q q0)
• FEM Nodal displacement X
• Vector of nodes displacement in global frame.
• Distance preserving transformation.
• X T (q - q0)

D- space
T
Physical Space
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Potential Energy

• Linear Elasticity.
• K FEM stiffness matrix. (2N ?? 2N matrix
  for N nodes.)
• Forces at nodes
• F K X.
• For FEM with linear elasticity and linear
  interpolation,
• U(q q0) (1/2) XT K X

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Equilibrium Deformations

• Equilibrium
• Local minimum of U.
• Stable equilibrium
• Strict local minimum of U.

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Workholding

• Part returns to original deformation stable
  equilibrium.
• Minimum work of UA required to release part.
• Caging grasps, saddle points Rimon99

UA
Returns to qA
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Deform Closure

• Stable equilibrium Deform Closure where
• UA gt 0.

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Theorem Frame Invariance

• Independence from global coordinate frame.
• Proved by showing invariance of
• - Deformation.
• - Potential energy and work.
• - Continuity in D-space.

M
E
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Theorem Equivalence
Form-closure of rigid part
Deform-closure of equivalent deformable part.
?
?
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Numerical Example
4 Joules
547 Joules
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Symmetry in D-Space

• D-Obstacle symmetry
• Obstacle identical for all mesh triangles.
• Prismatic extrusions.

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Symmetry in D-Space

• Topology preservation symmetry.
• Define D'T
• - No mesh collisions.
• - No degenerate triangles.
• DT ?? D'T.
• Mirror images
• - No continuous path.
• D'T identical for pairs of mesh triangles.

1
4
5
3
2
4
1
5
2
3
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Future work

• Optimal 2-finger deform closure
• Given jaw positions.
• Determine optimal jaw separation s .

s
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Quality Metric

• If Quality metric Q UA

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Quality metric

• Plastic deformation

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Quality metric
Stress
eL
Strain
Plastic Deformation
Q min UA, UL
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3D Meshes

• Tetrahedral elements
• - 3 DOF per node.
• Box elements
• - Translational Rotational DOF.
• Sheet metal
• - Shell elements.

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Contact Graph
Potential Energy
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Numerical Example
Undeformed s 10 mm.
Optimal se 5.6 mm.
Rubber foam. FEM performed using ANSYS.
Computing Deform Closure Grasps, K. "Gopal"
Gopalakrishnan and Ken Goldberg, submitted to
Workshop on Algorithmic Foundations of Robotics
(WAFR), Oct. 2004.
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Thank You
http//alpha.ieor.berkeley.edu