The non-overlapping constraint between objects described by non-linear inequalities

The non-overlapping constraint between objects described by non-linear inequalities

Ignacio SALAS & Gilles CHABERT & Alexandre GOLDSZTEJN

Content

Motivation

Conclusions

Introduction

The Algorithm

I. Salas & G. Chabert & A. Goldsztejn

Experimental Results

Overlapping as a Minkowski sum

Motivation

I. Salas & G. Chabert & A. Goldsztejn 1 / 23

Which are the positions and orientations of an object such that it does not overlap a second (fixed) one?

A key question in the packing problem

Introduction: The Object Definition


The object can be translated...

Where

…and also rotated

An object can be described by a constraint



● We consider objects that can be represented by a non-linear inequality

● And our results carry over the more general case of first order formulas

Disjunctions of conjunctions of inequalities



1 inequality Conjunction of inequalities

Disjunction of conjunctions of inequalities

Allows to consider the packing space (e.g. a bin) as a regular object

Introduction: The Non-Overlapping Constraint


“Reference”Object

(the fixed one)

“Moving”Object

(the variables)

Overlapping Region

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



Overlapping Constraint

The position of the moving object is the variable of the system

Considering the rotation angle



The overlapping region considering only translation, takes the following form

… and considering also rotation

Introduction: Paving

Outer Boxes

Inner Boxes

Boundary Boxes


Overlapping as a Minkowski Sum


What is the Minkowski Sum?

If the sets are described by constraints



Comparing

… and

… leads to



This is equal to the overlapping constraint


In the case of rotation, we consider “augmented” objects



The overlapping constraint S’ is still obtained as a Minkowski sum:

S’ I. Salas & G. Chabert & A. Goldsztejn 13 / 23

The Algorithm

Inner contraction

Outer rejection test

Bisection


The process stops when the surface of the unknown region B is less than % of the initial box [x]

● Our objective is to calculate a paving of the overlapping region

● our algorithm is based on a classical branch & prune algorithm and makes use of the Minkowski sum

The central operation can be broken into 3 steps:

[x]

The Algorithm: Inner Contractor


Given that

Let us consider

How can we find a subbox of [x] that is inside S?

Then

Before describing how to contract a box [x] ...

with



The resulting box is used in a sweep loop

must respect



The Sweep loopAt each step the point is “inflated” to the box

These boxes are pilled up until some face is covered

The Algorithm: Outer rejection test


If this assertion holds, the whole box is marked as outer

Moving ObjectReference Object

Experimental Results: Setup


● Variable ocurrences● Lack of convexity● Degrees of freedom

Difficulties?

3 objects of increasing difficulty

Each object can take 1 of the 2 roles

Reference Object

Moving Object

Experimental Results: Setup


Translation Translation & Rotation

We consider the 6 combinations of Reference-Moving objects

We consider 2 combinations (2 ellipsis and 2 peanuts)

2 types of experiments

Compared to what? RSolver

Experimental Results: Translation


Time vs Precision,

case 1

Time vs Precision,

case 6

RSolver Our Algorithm RSolver Our Algorithm

Obtained in (s.) 4.07 0.37 771.00 26.82

RSolver

Our Algorithm

= 3.25%

Experimental Results: Translation & Rotation


RSolver

Our Algorithm

Timeout after 80 minutes

Calculated in 9 minutes

Conclusions


We give an efficient way to calculate a guaranteed approximation of the non-overlapping constraint.

The efficiency is measured with respect to the time required for solving the quantified constraints directly.

In future works, we will try to solve the full packing problem using our results on the non-overlapping constraints.

The non-overlapping constraint between objects described by non-linear inequalities

Ignacio SALAS & Gilles CHABERT & Alexandre GOLDSZTEJN

Construction of an object

origin point

pp

We construct the object with respect the constraint, from the origin point

Obtaining the augmented set S’m

Conclusions: Future work

I. Salas & G. Chabert & A. Goldsztejn 23

We will try to solve the full packing problem using our results on the non-overlapping constraints.

Also, we want to know whether it is possible find a quick inner test.

The Object Definition

System U System U System U System1 2 3 4

The resulting shape is not necessarily convex

And this allows to represent the avalaible space for packing as a regular shape (called “enclosing shape”)

3

The Non-Overlapping Constraint

The overlapping with the enclosing shape is handled as with any other shape, since the description of the objects allows a transparent process.

4

yes

no no

yes

Between Regular Shapes Between an Enclosing Shape and a regular Shape

The Contractor: obtaining the forbidden box

Searching a point P

To find the point P, we use a branch & bound algorithm

12

Our objective is find a point that belongs to the moving object and to the compulsory part of the reference object


Searching a point P

Contract some box C w.r.t. the moving shape

Contract the box C w.r.t. the reference shape, centered in the lower left/upper right corner

13

In the beginning, it is an unbounded box

Removes points that are not in the compulsory part

Outer approximation of the moving object


Searching a point P

Pick randomly a point in the resulting box and check if it belongs to both objects

Else, BisectThe contracted box

Using interval evaluation

14


Inflate P

The point P is inflated using the box translated and the compulsory part

The inflators of Ibex can inflate inside inequalities

It is possible to inflate inside the same box all the inequalities that are part of the shape

15


Inflate P

For this we have defined an intersection of inner inflators ...

=

And a union of inner inflators

Choose the box with the greatest perimeter

16


Hence, it is possible to construct a box B inside the compulsory part of the reference object, i.e.

The Contractor: overlapping shape

● The overlapping shape is the region where the reference shape will always collide with the moving shape.

● Can be calculated exactly with polytopes, but we are using curved objects.

We calculate something different

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The Contractor: our forbidden box

Is an inner box of the reference shape

The intersection point

The forbidden box that we obtain belongs to this expression

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Inner Contractor

Reference Shape

Moving Shape

Domain of the origin of the moving shape (to be contracted)

● Target: Remove all the parts that violate the non-overlapping constraint

● The contraction operates over 2 shapes○ The reference shape○ The moving shape

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The Contractor

O1 ES O2 O3 O4

O2 ES O1 O3 O4

O3 ES O1 O2 O4

O4 ES O1 O2 O3

List of moving shapes

List of reference shapes

The origin box of each moving shape is contracted with Sweep.

Enclosing Shape

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Considering that the forbidden box must ● “extend” an initial forbidden point M● be included inside a “working area” (WA)


How to do it?

● Sweep requires the ability of calculating a forbidden box with respect to each constraint

● No algorithm exists so far for the non-overlapping constraint

The current domain in the context of the sweep loop

9

Satisfiability Check

To find this point, was used a branch & bound algorithmContract some box with the moving shape

Contract the reference shape centered in the some point

Evaluate the intersection in a random point of the resulting box

Else, Bisect

The satisfiability check works similar to the point selector, but only evaluate if the point exists

Sweep

● The target of the sweep algorithm is to prune all the parts of some domain that violate some constraint

● To perform this, we must “saturate” one of the dimensions with forbidden boxes

● This forbidden boxes must consider some forbidden point and the working area

The non-overlapping constraint between objects described by non-linear inequalities

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