K.

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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

• Summaries of results
• Mason, 2001
• Bicchi, Kumar, 2000
• Form and Force Closure
• Rimon, Burdick, 1995
• Rimon, Burdick, 1995
• Number of contacts
• Reuleaux, 1963, Somoff, 1900
• Mishra, Schwarz, Sharir, 1987, Markenscoff,
  1990
• Caging Grasps
• Rimon, Blake, 1999

Mason, 2001
3
Workholding Rigid parts

• Nguyen regions
• Nguyen, 1988
• Immobilizing three finger grasps
• Ponce, Burdick, Rimon, 1995
• 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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Mason, 2001
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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.

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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.

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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
• Manipulation of thin sheets
• - Kavraki et al, 1998

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Holding Deformable Parts

• Minimum Lifting Force
• - Howard, Bekey, 1999
• 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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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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D-Obstacles

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

Physical space
D-Space
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DTopological

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

Physical space
D-Space
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D-Space Example
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Self collisions and DTopological
Allowed deformation
Undeformed part
Topology violating deformation
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Free Space Dfree
1
4
5
2
3
Part and mesh
Slice with nodes 1-4 fixed
Slice with nodes 1,2,4,5 fixed
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Modeling Forces

• Nodal displacement X
• Vector of nodes displacement in global frame.
• Distance preserving transformation.
• X T (q - q0)
• Stiffness K
• F KX.
• Linear Elasticity.

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

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Potential Energy

• Nodal displacement
• Distance preserving transformation.
• X T (q - q0)

q0
Nominal mesh configuration

• For FEM with linear elasticity and linear
  interpolation,
• U(q q0) (1/2) XT K X

q
Deformed mesh configuration
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Equilibrium Deformations

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

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Releasing the Part.

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

UA
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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