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Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Motivation
• Simple reliable grips.
• Form-Closure achieved in 3D.
• Self-aligning grips.
• Very small Footprint.
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Related Work
• V-grips (Form-closure only in 2D).
[Gopalakrishnan, Goldberg, 2002]
• Multi DOF grippers in Robotic Fixtureless
Assembly.[Plut, Bone, 1997]
Related Work
• Form-Closure & Force Closure
– [Mason, 2001]
– [Rimon, Burdick, 1995 & 1996]
• Necessary & Sufficient Conditions (number of
contacts)
– [Realeaux, 1963]
– [Somoff, 1900]
– [Mishra, Schwarz, Sharir, 1987]
– [Markenscoff, 1990]
Related Work
Caging Grasps [Rimon, Blake, 1999]
Efficient Computation of Nguyen regions [Van der Stappen, Wentink, Overmars, 1999]
Multi-DOF Grips for Robotic Fixtureless Assembly [Plut, Bone, 1996 & 1997]
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Problem Definition
We first analyze piecewise planar sheet metal parts.
Assumptions:
• Frictionless contacts.
• Linear part edges (perimeter & holes).
• First order form-closure only.
• Jaws are vertical parallel cylinders with horizontal V-shaped grooves.
• Part thickness negligible.
V-shaped Grooves
• Consists of intersection of 2 frustums as shown:
• Forces exerted normal to each frustum at point of contact.
Jaw
Part
Analysis of Free Motion
• Any motion can be broken down to component
motions:
• Translations: ex, ey, ez
• Rotations: rx, ry, rz
• Any infinitesimal motion = sequence of
components.
Coordinate System
• x-z plane contains axes of jaws.
• Jaws close along x.
• Jaw axes parallel to z.
x
z y
Results from V-grips
Step1: We consider a pair of concave vertices.
Step2: At these vertices, we draw normalsto the edges through the jaw’s center.
Step3: We label the 4 regions as shown:
I
II
IV
III
Theorem:
Both jaws lie strictly in the other’s Region I means it is an expanding v-grip
orBoth jaws lie strictly in the other’s Region IV
means it is a contracting v-grip.
Form-Closure in the x-y plane
• Test of 2D v-grip is applied.
• Ensures that distance decreases for ex, ey, rz.
• Next, we consider ex, ez, ry.
x
z y
Form-Closure in the x-z plane
• Test of 2D v-grip is applied in x-z plane.
• However, consider the jaws as the part and part as
contacts.
• Ensures that distance decreases for ex, ez, ry.
• For infinitesimal motions, vertices still lie strictly
in corresponding region.
x
z y
Rotation about x axis
• Rotation about x axis always increases.
• Reason: Only parts which are horizontal at
contacts are considered.
Theorem
• Any sheet-metal part which is horizontal at the
points of contact is held in Form-Closure if
– Part is in expanding/contracting 2D v-grip
for horizontal projection.
– Jaws are held in contracting/expanding 2D
v-grip by the part in the plane containing
axes of jaws.
• These conditions are sufficient but not necessary.
Algorithm
• For every face of part, generate all pairs of
concave vertices on faces parallel to it.
• Test each pair for Form-Closure.
• If in Form-Closure, add to list of grips.
• Sort list by quality metric.
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Quality Metric for v-grips
• Based on sensitivity to relaxing of jaws.
Maximum change in orientation with one jaw still at a vertex.
|d/dl| = |tan()/l|
l
l-l
v a v b
Suggested Metric
• Sensitivity of orientation to relaxing of jaws.
• Consider all values of |tan()/l| for v-grips in x-y
and x-z planes.
• Take maximum value for worst change in
orientation.
• Intuitive but not rigorous.
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Diagonal Planes at Contacts
• Horizontality is used only when analyzing rx.
• If distance can be shown to reduce otherwise,
horizontal assumption can be removed.
• E.g. Plane contains a line parallel to y-axis.
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work
Strength of Conditions
• Compare results generated by sufficient conditions
with those generated by brute force.
lh
Comparison
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25
Min
imum
hal
f-v
angl
e of
fru
stum
l
Sufficient Conditions
Brute Force
ComparisonM
inim
um h
alf-
v an
gle
of f
rust
um
h
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25
Brute Force
Sufficient Conditions
Outline
• Introduction and Motivation
• Related Work
• Gripping Sheet Metal Parts
• Quality Metric
• Extensions
• Analysis of Results
• Conclusions & Future Work