Download - On the Existence of Form-Closure
![Page 1: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/1.jpg)
1
On the Existence of Form-Closure Configurations on a Grid
A. Frank van der Stappen
Presented by K. Gopalakrishnan
![Page 2: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/2.jpg)
2
• Inspiration
• Introduction
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
![Page 3: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/3.jpg)
3
• Modular fixturing
• Existence of Fixtures– [Zhuang, Goldberg, 96]
• Purely geometric approach
Inspiration
![Page 4: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/4.jpg)
4
• Inspiration
• Introduction & Related Work
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
![Page 5: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/5.jpg)
5
Ways to hold parts
• Form Closure– Any part motion causes
collision
• Force Closure– Any external Wrench resisted
by applying suitable forces
[Mason, 2001]
![Page 6: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/6.jpg)
6
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)
![Page 7: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/7.jpg)
7
ObstaclesObstacles prevent parts from moving freely.
These images of obstacles in C-space are called C-obstacles.
The rest is Free Space.
![Page 8: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/8.jpg)
8
Form Closure
Form Closure:
All adjacent points are collisions
Difficult to Determine.Hence, distance from C-obstacles is used. [Rimon & Burdick, 96]
![Page 9: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/9.jpg)
9
Form Closure in C-space
“Active” obstacles only
Small motion cannot result in reduction of distance.
Distance not easy to compute either.
![Page 10: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/10.jpg)
10
First order Form-Closure• Consider Infinitesimal motion.
• Taylor Expansion for distance.
• Truncate to First order.
• Equivalent to replacing surfaces by tangents.
![Page 11: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/11.jpg)
11
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]
![Page 12: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/12.jpg)
12
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.
![Page 13: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/13.jpg)
13
Fast Test
+ -
![Page 14: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/14.jpg)
14
Fast Test
+ -
-+
![Page 15: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/15.jpg)
15
Fast Test
+ -
-+
![Page 16: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/16.jpg)
16
Fast Test
+ -
-+
++-
-
![Page 17: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/17.jpg)
17
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]
![Page 18: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/18.jpg)
18
Second Order Form-Closure
Example: 1st order approximation:
![Page 19: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/19.jpg)
19
Second order Form-Closure
First order approximation allows a pure rotation about the centroid.
![Page 20: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/20.jpg)
20
• Inspiration
• Introduction
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
![Page 21: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/21.jpg)
21
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.
![Page 22: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/22.jpg)
22
Notation
• P Polygonal Part
• e any edge of P
• a any point on e
• le(a) normal to e at a
![Page 23: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/23.jpg)
23
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
![Page 24: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/24.jpg)
24
Proof
• 3 radial vectors positively span R2.
• m only possible center of rotation.
• Contacts obtained in neighborhood of ai.
a1 a2
a3
m
![Page 25: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/25.jpg)
25
Case I
• At least 1 ai is a vertex.
• Has to be Concave.
a1 a2
a3
m
![Page 26: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/26.jpg)
26
Case I
• If no normal at vertex passes thru m, we are done
a1 a2
a3
m
![Page 27: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/27.jpg)
27
Case I
• If a normal at the vertex passes through m,
a1 a2
a3
m
![Page 28: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/28.jpg)
28
Case I
![Page 29: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/29.jpg)
29
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.
![Page 30: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/30.jpg)
30
Case II-1
• m is to the left of normal at a4.
• Move the horizontal line slightly upward.
![Page 31: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/31.jpg)
31
Case II-2
• m is to the right of normal at a4.
• Rotate both axes by equal angles.
![Page 32: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/32.jpg)
32
Case II-3
• m is to on the normal at a4.
• Coordinated translation of horizontal and vertical gridlines.
+
-
+ -
+
-
+ -
![Page 33: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/33.jpg)
33
• Inspiration
• Introduction
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
![Page 34: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/34.jpg)
34
Parallel Edges
• Counterexamples.
• Topology not maintained.
![Page 35: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/35.jpg)
35
Contact at Grid Vertex
• Not feasible (e.g. convex polygons).
• Additional contact on another line required.
• Feasibility not guaranteed.
• Requires large enough part.
![Page 36: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/36.jpg)
36
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.
![Page 37: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/37.jpg)
37
Jaws with non-zero radii
• Contact at Convex vertex not desired.
• As radius increases, Minkowsky sum becomes disc.
![Page 38: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/38.jpg)
38
• Inspiration
• Introduction
• Form Closure on a Grid
• Extensions
• Conclusion
Outline
![Page 39: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/39.jpg)
39
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.
![Page 40: On the Existence of Form-Closure](https://reader035.vdocument.in/reader035/viewer/2022062406/558b4c6fd8b42a342a8b4702/html5/thumbnails/40.jpg)
40