Deterministic Pattern Formation in Swarms of Robots

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Deterministic Pattern Formation in Swarms of Robots

Franck Petit

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Problem

Definition (Arbitrary Pattern Formation)

Given a swarm ofnrobots scat- tered on the plan, designing a deterministic algorithm so that, the robots eventually form a pat- ternP made of n positions and known by each of them in ad- vance.

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Problem

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Problem

In other words, at the end of the computation, the positions of the robots coin- cide, with the positions ofP, wherePcan be translated, rotated, and scaled in each local coordinate system.

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Previous Works

SoD and Chirality

[Flocchini et al., 1999]

A solution exists for any n and anyP.

SoD and No Chirality

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Previous Works

SoD and Chirality

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Previous Works

SoD and Chirality

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Previous Works

SoD and Chirality

Ifnis odd, a solution exists for anyP.

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Previous Works

SoD and Chirality

Ifnis odd, a solution exists for anyP.

If nis even, a solution exists provided thatPissymmetric.

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Pattern Formation With No Sense of Direction

Question

Assuming no sense of direction (with or without chirality), which kind of patterns can be formed in adeterministicway?

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Pattern Formation With No Sense of Direction

Theorem

If the robots shareno Sense of Direction, then they cannot solve the Arbitrary Pattern Formation problemdeterministically, even having the ability of chirality.

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Pattern Formation With No Sense of Direction

Theorem

x

δ δ

δ

δ δ δ

δ δ

y x

y

y y y

y

x

x x x x

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Pattern Formation With No Sense of Direction

Theorem

LetΘbe the class of patterns that can be solved

deterministicallyassuming robots devoid of sense of direction.

Corollary

Either Θis equal to the set of regular polygons (n-gons)

or Θ =∅.

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Circle Formation

Question

Assuming no sense of direction, is it possible to eventually form a regularn-gon (circle)?

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Circle Formation

Previous work

[D ´efago and Konagaya, 2002] [Chatzigiannakis et al., 2004] [D ´efago and Souissi, 2008]

Asymptotic convergencetoward then-gon.

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Circle Formation

Previous work

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Circle Formation

Previous work

[Dieudonn ´e and Petit, 2007]

Primenumber of robots.

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Circle Formation

Previous work

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Circle Formation

Previous work

[Katreniak, 2005]

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Circle Formation

Previous work

[Katreniak, 2005]

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Circle Formation

All previous protocols are based on the two following steps:

1 Move the robots on (the boundary of) a circleC.

2 Without leavingC, arrange them evenly alongC.

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Circle Formation

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Circle Formation

δ

δ δ

δ

δ δ

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Circle Formation

EASY PART

δ

δ δ

δ

δ δ δ

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Circle Formation

EASY PART

δ

δ δ

δ

δ δ

HARD PART

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Circle Formation

Theorem

[Flocchini et al, 2006]

There existsno deterministic algorithm that eventually arrange n robots evenly along a circleCwithout leavingC.

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Circle Formation

Theorem

α α

α

α β

β

β β

δ

δ δ

δ

δ δ δ

IMPOSSIBLE

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Circle Formation

Theorem

α α

α

α β

β

β β

δ

δ δ

δ

δ δ δ

IMPOSSIBLE

Question

Are we done,i.e.,Θ =∅?

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Circle Formation

Theorem

α α

α

α β

β

β β

δ

δ δ

δ

δ δ δ

IMPOSSIBLE

Question

Are we done,i.e.,Θ =∅?

Fortunately, not!

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An almost n-gon algorithm

Arbitrary Configuration

Biangular Circle

Regular Polygon

[Katreniak, 2005]

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Starting from a biangular circle

Theorem

There existsno deterministic algorithm that eventually arrange n robots evenly along a circleCwithout leavingC.

α α

α

β β

δ

δ δ

δ

δ δ δ

IMPOSSIBLE

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Starting from a biangular circle

Let us leaveC

α α

β

α β

β

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Starting from a biangular circle

Let us leaveC

α α

β

α β

β

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Starting from a biangular circle

Let us leaveC

α α

β

α β

β

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Starting from a biangular circle

Let us leaveC

α α

β

α β

β

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Starting from a biangular circle

Let us leaveC Everyrobot is active

Regular Polygon

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Starting from a biangular circle

Let us leaveC Everyrobot is active

Regular Polygon

Somerobots are active

Ideal Polygon

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Starting from a biangular circle

Ideal Polygon

d

d d

2

3

4 5

0

1

Property 1

Eitherd0≥d1≤ · · · ≥dn−1≤d0

ord0≤d1≥ · · · ≤dn−1≥d0

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Starting from a biangular circle

Ideal Polygon

Property 2

A regular n-gon can be associated to an Ideal Polygon.

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Starting from a biangular circle

Ideal Polygon

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Circle Formation

Arbitrary Configuration

Biangular Circle

Regular Polygon

Ideal Polygon

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Circle Formation

α

β β

It works ifn≥5 only

1 [Katreniak 2005] works if n ≥ 5 only

2 Difficulty to find an invariant

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(nR−P) Non−Rectangle Parallelogram

e

(nR−IT) Non−Rectangle Isosceles Trapezoid (AQ) Asymetric Quadrilateral

(nD−T) Non−Delta Triangle

q r s

t L’

L (nP−D) Non−Perpendicular Delta

q r s

t

L (P−D) Perpendicular Delta

(nP−R) Non−Perpendicular Rectangle

dmax O

(PQ) Perpendicular Quadrilateral

O

O (sL) The robots are aligned

L

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Arbitrary Pattern Formation

0000 1111

etc...

Theorem

∀n, the class of patterns that can be solved deterministically assuming robots devoid of sense of direction (Θ) is equal to the set of regular polygons (n-gons).