The document discusses the non-overlapping constraint for objects described by non-linear inequalities, focusing on methods for determining object positions and orientations to avoid overlap. It introduces an algorithm based on a branch and prune approach, employing Minkowski sums to analyze overlapping regions. Experimental results indicate the algorithm's efficiency and its potential for further developments in solving complex packing problems.

1. The non-overlapping
constraint between
objects described by
non-linear
inequalities
Ignacio SALAS & Gilles CHABERT &
Alexandre GOLDSZTEJN

2. Content
Motivation
Conclusions
Introduction
The Algorithm
I. Salas & G. Chabert & A. Goldsztejn
Experimental Results
Overlapping as a Minkowski sum

3. 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

4. Introduction: The Object Definition
I. Salas & G. Chabert & A. Goldsztejn 2 / 23
The object can
be translated...
Where
…and also
rotated
An object can be
described by a constraint

5. Introduction: The Object Definition
I. Salas & G. Chabert & A. Goldsztejn 3 / 23
● 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

6. Introduction: The Object Definition
I. Salas & G. Chabert & A. Goldsztejn 4 / 23
1 inequality Conjunction of
inequalities
Disjunction of
conjunctions of
inequalities
Allows to consider the
packing space (e.g. a
bin) as a regular object

7. Introduction: The Non-Overlapping
Constraint
I. Salas & G. Chabert & A. Goldsztejn 5 / 23
“Reference”
Object
(the fixed one)
“Moving”
Object
(the variables)
Overlapping
Region
The non-overlapping constraint is the negation of the
overlapping constraint

8. Introduction: The Non-Overlapping
Constraint
I. Salas & G. Chabert & A. Goldsztejn 6 / 23
Overlapping Constraint
The position of the moving object is the variable of the
system
Considering the rotation
angle

9. Introduction: The Non-Overlapping
Constraint
I. Salas & G. Chabert & A. Goldsztejn 7 / 23
The overlapping region considering only translation,
takes the following form
… and considering also rotation

10. Introduction: Paving
Outer Boxes
Inner Boxes
Boundary
Boxes
I. Salas & G. Chabert & A. Goldsztejn 8 / 23

11. Overlapping as a Minkowski Sum
I. Salas & G. Chabert & A. Goldsztejn 9 / 23
What is the Minkowski Sum?
If the sets are described by constraints

12. Overlapping as a Minkowski Sum
I. Salas & G. Chabert & A. Goldsztejn 10 / 23
Comparing
… and
… leads to

13. Overlapping as a Minkowski Sum
I. Salas & G. Chabert & A. Goldsztejn 11 / 23
This is equal to the
overlapping constraint

14. Overlapping as a Minkowski Sum
In the case of rotation, we consider
“augmented” objects
I. Salas & G. Chabert & A. Goldsztejn 12 / 23

15. Overlapping as a Minkowski Sum
The overlapping constraint S’ is still obtained as
a Minkowski sum:
S’
I. Salas & G. Chabert & A. Goldsztejn 13 / 23

16. The Algorithm
Inner contraction
Outer rejection test
Bisection
I. Salas & G. Chabert & A. Goldsztejn 14 / 23
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]

17. The Algorithm: Inner Contractor
I. Salas & G. Chabert & A. Goldsztejn 15 / 23
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

18. The Algorithm: Inner Contractor
I. Salas & G. Chabert & A. Goldsztejn 16 / 23
The resulting box is
used in a sweep loop
must respect

19. The Algorithm: Inner Contractor
I. Salas & G. Chabert & A. Goldsztejn 17 / 23
The Sweep loop
At each step the
point is
“inflated” to the box
These boxes are
pilled up until some
face is covered

20. The Algorithm: Outer rejection test
I. Salas & G. Chabert & A. Goldsztejn 18 / 23
If this assertion holds, the whole
box is marked as outer
Moving Object
Reference Object

21. Experimental Results: Setup
I. Salas & G. Chabert & A. Goldsztejn 19 / 23
● 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

22. Experimental Results: Setup
I. Salas & G. Chabert & A. Goldsztejn 20 / 23
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

23. Experimental Results: Translation
I. Salas & G. Chabert & A. Goldsztejn 21 / 23
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%

24. Experimental Results: Translation &
Rotation
I. Salas & G. Chabert & A. Goldsztejn 22 / 23
RSolver
Our
Algorithm
Timeout after 80
minutes
Calculated in 9
minutes

25. Conclusions
I. Salas & G. Chabert & A. Goldsztejn 23 / 23
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.

26. ??

27. The non-overlapping
constraint between
objects described by
non-linear
inequalities
Ignacio SALAS & Gilles CHABERT &
Alexandre GOLDSZTEJN

28. Construction of an object
origin point
p
p
We construct the object with
respect the constraint, from
the origin point

29. Obtaining the augmented set S’m

30. 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.

31. 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

32. 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

33. 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

34. The Contractor:
obtaining the forbidden box
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

35. The Contractor:
obtaining the forbidden box
Searching a point P
Pick randomly a point in the resulting box and
check if it belongs to both objects
Else, Bisect
The contracted box
Using interval
evaluation
14

36. The Contractor:
obtaining the forbidden box
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

37. The Contractor:
obtaining the forbidden box
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

38. The Contractor:
obtaining the forbidden box
Hence, it is possible to construct
a box B inside the compulsory
part of the reference object, i.e.

39. 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
17

40. 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
18

41. 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
7

42. 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
8

43. Considering that the forbidden box must
● “extend” an initial forbidden point M
● be included inside a “working area” (WA)
The Contractor:
obtaining the forbidden box
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

44. Satisfiability Check
To find this point, was used a branch & bound algorithm
Contract 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

45. 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