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

5

Definitions: The non-overlapping constraint

The non-overlapping constraint is the negation of the latter relation

6

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

7

Definitions: Overlapping FunctionThe overlapping function must fulfill two properties:

8

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:

9

Definitions: Overlapping Region

Overlapping Region

10

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

11

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

13

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

15

Cartesian Product [ᾱ , ⍶][oj]

Experimental Results

16

Off-line

On-line

5 experimental cases

Circle Packing

Ellipse Packing

Ellipse Packing + rotation

Horseshoes Packing

Mixed Packing

Case 4

Case 5

1

2

3

4

5

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.

17

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