Title: K.
1 D-Space and Deform Closure A Framework
for Holding Deformable Parts
- K. Gopal Gopalakrishnan, Ken Goldberg
- IEOR and EECS, U.C. Berkeley.
2Workholding 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
3Workholding 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
4C-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.
5Avoiding Collisions C-obstacles
- Obstacles prevent parts from moving freely.
- Images in C-space are called C-obstacles.
- Rest is Cfree.
6Workholding 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
7Form 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
8Holding 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
9Holding Deformable Parts
- Minimum Lifting Force
- - Howard, Bekey, 1999
- Quasi-static path planning.
- - Anshelevich et al, 2000
- Robust manipulation
- - Wada, Hirai, Mori, Kawamura, 2001
10Deformable parts
- Form closure does not apply
- Can always avoid collisions by deforming the part.
11D-Space
- Deformation Space A Generalization of
Configuration Space. - Based on Finite Element Mesh.
12Deformable Polygonal parts Mesh
- Planar Part represented as Planar Mesh.
- Mesh nodes edges Triangular elements.
- N nodes
- Polygonal boundary.
13D-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
14D-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
15D-Obstacles
- Collision of any mesh triangle with an object.
- Physical obstacle Ai has an image DAi in D-Space.
Physical space
D-Space
16DTopological
- Mesh topology maintained.
- Non-degenerate triangles only.
Physical space
D-Space
17D-Space Example
18Self collisions and DTopological
Allowed deformation
Undeformed part
Topology violating deformation
19Free Space Dfree
1
4
5
2
3
Part and mesh
Slice with nodes 1-4 fixed
Slice with nodes 1,2,4,5 fixed
20Modeling 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)
21Potential 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
22Equilibrium Deformations
- Equilibrium
- Local minimum of U.
- Stable equilibrium
- Strict local minimum of U.
23Releasing the Part.
- Part returns to original deformation.
- Minimum work of UA required to release part.
- Caging grasps, saddle points Rimon99
UA
24Deform Closure
- Stable equilibrium Deform Closure where
- UA gt 0.
25Theorem Frame Invariance
- Independence from global coordinate frame.
- Proved by showing invariance of
- - Deformation.
- - Potential energy and work.
- - Continuity in D-space.
M
E
26Theorem Equivalence
Form-closure of rigid part
Deform-closure of equivalent deformable part.
?
?
27Numerical Example
4 Joules
547 Joules
28Symmetry in D-Space
- D-Obstacle symmetry
- Obstacle identical for all mesh triangles.
- Prismatic extrusions.
29Symmetry 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
30Future work
- Optimal 2-finger deform closure
- Given jaw positions.
- Determine optimal jaw separation s .
s
31Quality Metric
32Quality metric
33Quality metric
Stress
eL
Strain
Plastic Deformation
Q min UA, UL
343D Meshes
- Tetrahedral elements
- - 3 DOF per node.
- Box elements
- - Translational Rotational DOF.
- Sheet metal
- - Shell elements.
35Contact Graph
Potential Energy
36Numerical 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.
37Thank You
http//alpha.ieor.berkeley.edu