Packing Curved Objects
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Transcript of Packing Curved Objects
Packing Curved ObjectsIgnacio SALAS & Gilles CHABERT
OutlineMotivation
Method
Method: Inner InflatorExperimental Results
Conclusions
Definitions
2
Motivation
3
What is the packing problem?Well studied with :
Circles Bins
We deal with the more general case where different shapes can be mixed
Including non-convex objects and curved shapes
Non-Overlapping Constraint
Motivation: Using CMA-ESIn [Mar13] the packing problem was solved minimizing a violation function with the CMA-ES algorithm. The function is a measure of overlapping.
The approach gives encouraging results, but requires ad-hoc distance functions for each pair of objects
Our objective is to replace these ad-hoc formulas by a numerical algorithm4
Definitions: The objects
We consider objects described by nonlinear inequalities,with and/or operators in the shape description
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Definitions: The non-overlapping constraint
The non-overlapping constraint is the negation of the latter relation
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qi and qj are parameters of Objects n°i and j
Rotation Translation
Definitions: The packing problem
The packing problem is therefore a set of pairwise non-overlapping constraints between n+1 objects
n obects to pack inside a container space
Inclusion in the container can also be seen as a non-overlapping constraint
We consider the complementary of the container
c1
c2
c3
¬c
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Definitions: Overlapping FunctionThe overlapping function must fulfill two properties:
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Decreases when the objects get more distant
Takes the value of 0 if the objects are disjoint
Definitions: Overlapping Function
A ⦅distance to satisfaction⦆
A generic definition is:
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Definitions: Overlapping Region
Overlapping Region
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where
Which is the relation between the overlapping function and the overlapping region?
Off-line
Method
Calculate the Overlapping Region Sij
Calculate the minimal distance between r(qi, qj) and Sij
Paving Algorithm
Paving of the Overlapping Region
Distance calculation
qi
qj
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Object i
Object j
Method: Paving of the Overlapping Region
Outer Rejection Test
Inner Inflation [detailed further]
Bisection
Branch and Bound algorithm, that alternates 3 steps:
Starts with an arbitrary large box [qj]
The paving is stored in a tree structure12
Method: Distance to the boundary setFind the closest point q’’ that does not belongs to the overlapping region
The distance to q’’ is in fact an interval:
The boxes in O are scanned in logarithmic time
Thanks to the tree-structure representation
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Resulting inflation
Method: Inner Inflation
p satisfies14
Translation Rotation
The boundary angles of [ᾱ , ⍶] are two angles that makes the boundary of Object j meets p
~~
Method: Inner Inflation
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Cartesian Product [ᾱ , ⍶][oj]
Experimental Results
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Off-line
On-line
5 experimental cases
Circle Packing
Ellipse Packing
Ellipse Packing + rotation
Horseshoes Packing
Mixed Packing
Case 4
Case 5
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2
3
4
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Conclusions
We have presented a numerical algorithm that replaces such formulas.
The experimental results shows that our approach:
In [Mar13] was proposed an original approach for solving the generic packing problem, requiring ad-hoc distance formulas.
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Is not competitive for standard packing problems.
But is able to pack arbitrary objects, including non-convex ones.
The approach is particularly well-suited for uniform packing.
Limitation: the processing time increases with the number of shapes.
Objects with the same shape
Thank You !18
Packing Curved ObjectsIgnacio SALAS & Gilles CHABERT