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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]

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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]

[Mason, 2001]

+ -

+-

++-

-

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C-Space

C-Space (Configuration Space):

• [Lozano-P’erez, 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

q

4

5

/3(5,4,- /3)

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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.

PartObstacle

PartObstacle

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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.

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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.

Part

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• Bounded force-closure

- [Wakamatsu, Hirai, Iwata, 1996]

• Manipulation of thin sheets

- [Kavraki et al, 1998.]• Robust manipulation

- [Wada, Hirai, Mori, Kawamura, 2001]

Holding Deformable Parts

[Wada et al]

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Deformable parts

• “Form closure” does not apply:

Can always avoid collisions by deforming the part.

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• Deformation Space: A Generalization of Configuration Space.

• Based on Finite Element Mesh.

D-Space

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Mesh M

Part E

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.

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D-Space: Example• Simple example:

4-noded mesh.• D-Space: 8-dimensional Euclidean Space.• 2D slices show each mesh node’s position.• Node positions also indicate part orientation.

1 3

42

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D-Obstacles

No collision

Collision

Collision

1 3

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Slice of complement of

D-obstacle (DAi).Nodes 1,2,3 fixed.

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Topology violating

deformation

Undeformed part

Allowed deformation

Self collisions and DTopological

TDq

TDq

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Free Space: Dfree

Slice with nodes 1-4 fixed

Part and mesh

1

2 3

5

4

x

y

Slice with nodes 1,2,4,5 fixed

x3

y3x5

y5

x5

y5

x5

y5

i

iTfree DADD

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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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Nominal mesh configuration

Deformed mesh configuration

Potential Energy

• Nodal displacement:

Distance preserving transformation.

X = T (q - q0)

q0

q

• 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.

qA

qB

q

U(q)

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Releasing the Part.

• Part should slide back to original deformation.

• Minimum work of UA needs to be done to release part.

• Caging grasps, saddle points [Rimon99]

qA

qB

q

U(q)

UA

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Deform Closure

• Stable equilibrium = Deform Closure where

• UA > 0.

qA

qB

q

U(q)

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• Independence from global coordinate frame.

• Proved by showing invariance of:

- Deformation.

- Potential energy and work.

- Continuity in D-space.

Theorem: Frame Invariance

M

E

x1

y1

x 2

y 2

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Form-closure of rigid part

Theorem: Equivalence

Deform-closure of equivalent deformable part.

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Numerical Example

4 Joules 547 Joules

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• D-Obstacle symmetry

• Obstacle identical for all mesh triangles.

• Prismatic extrusions.

Symmetry in D-Space

1

32

4

5

1

32

4

5

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• 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.

Symmetry in D-Space

1

32

4

5

4

23

1

5

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Optimal 2-finger deform closure:

• Given jaw positions.

• Determine optimal jaw separation *.

Future work

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• If Quality metric Q = UA:

Quality Metric

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Quality metric

• Plastic deformation:

30Q = min { UA, UL }

Stress

Strain

Plastic Deformation

L

Quality metric

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Holding multiple parts:

• Fixturing sheet metal parts for welding.

• Relative displacements of nodes.

• Quadratic programming approach.

Future work

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