1 on the existence of form- closure configurations on a grid a.frank van der stappen presented by k....
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1
On the Existence of Form-Closure Configurations on a Grid
A. Frank van der Stappen
Presented by K. Gopalakrishnan
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• Modular fixturing
• Existence of Fixtures– [Zhuang, Goldberg, 96]
• Purely geometric approach
Inspiration
4
• Inspiration
• Introduction & Related Work
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
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Ways to hold parts
• Form Closure– Any part motion causes
collision
• Force Closure– Any external Wrench resisted
by applying suitable forces
[Mason, 2001]
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C-Space
C-Space (Configuration Space)Describes position and orientation.
We represent each degree of freedom of a part as a C-space coordinate.
y
x
/3
(5,4)
y
x
4
5
/3(5,4,-/3)
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ObstaclesObstacles prevent parts from moving freely.
These images of obstacles in C-space are called C-obstacles.
The rest is Free Space.
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Form Closure
Form Closure:
All adjacent points are collisions
Difficult to Determine.Hence, distance from C-obstacles is used. [Rimon & Burdick, 96]
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Form Closure in C-space
“Active” obstacles only
Small motion cannot result in reduction of distance.
Distance not easy to compute either.
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First order Form-Closure• Consider Infinitesimal motion.
• Taylor Expansion for distance.
• Truncate to First order.
• Equivalent to replacing surfaces by tangents.
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First order Form-Closure
In n dimensions there are n(n+1)/2 DOF. n translations n(n-1)/2 rotations
For first order form-closure, n(n+1)/2+1 are necessary and sufficient– [Realeaux, 1963]
– [Somoff, 1900]
– [Mishra, Schwarz, Sharir, 1987]
– [Markenscoff, 1990]
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Fast Test for First OrderForm-Closure
• Any infinitesimal motion on the plane is a rotation.
• No center of rotation possible for a part in Form-Closure.
• Try to identify possible centers.
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Failure of First order Form-Closure
Higher order terms neglected.
Uncertainty when first order term is 0.
We need to look at Second order terms.
For generic parts in 2 or 3 dimensions, 3 or 4 point contacts are sufficient. [Rimon, Burdick, 1995]
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Problem Definition
• Rigid polygonal part.
• Frictionless point contacts.
• No parallel edges.
To prove that:
The part can be held in form-closure by 4 point-contacts that lie on 2 perpendicular lines.
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Proof
• Consider the largest inscribed circle C in P.
• C has center m
• Let this touch the part at a1, a2, a3.
a1 a2
a3
m
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Proof
• 3 radial vectors positively span R2.
• m only possible center of rotation.
• Contacts obtained in neighborhood of ai.
a1 a2
a3
m
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Case II
• All 3 contacts at edges.
• Again, m is the only possible
• Let the least angle between radii be between ma1 and ma3.
• We choose a4 to make a1a4 and a2a3 gridlines.
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Case II-3
• m is to on the normal at a4.
• Coordinated translation of horizontal and vertical gridlines.
+
-
+ -
+
-
+ -
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Contact at Grid Vertex
• Not feasible (e.g. convex polygons).
• Additional contact on another line required.
• Feasibility not guaranteed.
• Requires large enough part.
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Jaws with non-zero radii
Jaw has a radius r
The part is transformed with a Minkowsky addition, offsetting the polygons with a disk of radius r.
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Jaws with non-zero radii
• Contact at Convex vertex not desired.
• As radius increases, Minkowsky sum becomes disc.
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Conclusion
• Existence of solution proved under assumptions.
• Validity of assumptions shown.
• Non zero radius incorporated.
• Extension to non-polygonal parts open.
• Accessibility to clamps ignored.