Square Formation by Asynchronous Oblivious Robots CCCG 2016 - - PowerPoint PPT Presentation

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Square Formation by Asynchronous Oblivious Robots CCCG 2016 - - PowerPoint PPT Presentation

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Square Formation by Asynchronous Oblivious Robots CCCG 2016 Marcello Mamino, Giovanni Viglietta Vancouver August 3, 2016 Square Formation by Asynchronous Oblivious Robots Anonymous robots sensing and moving in the plane We consider a

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Square Formation by Asynchronous Oblivious Robots

CCCG 2016 Marcello Mamino, Giovanni Viglietta Vancouver – August 3, 2016

Square Formation by Asynchronous Oblivious Robots

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Anonymous robots sensing and moving in the plane

We consider a swarm of anonymous robots in the Euclidean plane

Square Formation by Asynchronous Oblivious Robots

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Anonymous robots sensing and moving in the plane

Each robot can sense the positions of all other robots...

Square Formation by Asynchronous Oblivious Robots

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Anonymous robots sensing and moving in the plane

Each robot can sense the positions of all other robots...

Square Formation by Asynchronous Oblivious Robots

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Anonymous robots sensing and moving in the plane

Each robot can sense the positions of all other robots...

Square Formation by Asynchronous Oblivious Robots

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Anonymous robots sensing and moving in the plane

Each robot can sense the positions of all other robots...

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Anonymous robots sensing and moving in the plane

...And move according to a deterministic algorithm

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Anonymous robots sensing and moving in the plane

...And move according to a deterministic algorithm

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Anonymous robots sensing and moving in the plane

Different robots are activated asynchronously

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Anonymous robots sensing and moving in the plane

Different robots are activated asynchronously

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Anonymous robots sensing and moving in the plane

Different robots are activated asynchronously

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Anonymous robots sensing and moving in the plane

Different robots are activated asynchronously

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

Problem: form a given pattern from any initial configuration

Square Formation by Asynchronous Oblivious Robots

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Pattern Formation problem

The pattern may be rotated, reflected, and scaled

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Pattern Formation problem

The pattern may be rotated, reflected, and scaled

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Pattern Formation problem

The pattern may be rotated, reflected, and scaled

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Pattern Formation problem

The pattern may be rotated, reflected, and scaled

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Model definition

Robots are: Dimensionless (robots are modeled as geometric points) Anonymous (no unique identifiers) Homogeneous (the same algorithm is executed by all robots) Autonomous (no centralized control) Oblivious (no memory of past events) Silent (no explicit way of communicating) Long-sighted (complete visibility of all other robots) Disoriented (robots do not share a common reference frame, and a robot’s reference frame may change from turn to turn)

No common unit distance No common compass No common notion of clockwise direction

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Life cycles and asynchronicity

Look / Compute Move

Each robot repeats a Look/Compute/Move cycle

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Life cycles and asynchronicity

Look / Compute Move

Each robot repeats a Look/Compute/Move cycle

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Life cycles and asynchronicity

Look / Compute Move

/In a Look phase, an instantaneous snapshot is taken of all robots/

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Life cycles and asynchronicity

Look / Compute Move

/A destination point is computed as a function of the snapshot/

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Life cycles and asynchronicity

Look / Compute Move

/The destination point is approached with unpredictable speed/

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Life cycles and asynchronicity

Look / Compute Move

/The destination point is approached with unpredictable speed/

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Life cycles and asynchronicity

Look / Compute Move

/The destination point is approached with unpredictable speed/

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Life cycles and asynchronicity

Look / Compute Move

/The destination point is approached with unpredictable speed/

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Life cycles and asynchronicity

Look / Compute Move

/The destination point is approached with unpredictable speed/

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Life cycles and asynchronicity

Look / Compute Move

The robot may unpredictably/ stop before reaching the destination...

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Life cycles and asynchronicity

Look / Compute Move

...and execute a new Look/Compute phase

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Life cycles and asynchronicity

Look / Compute Move

...and execute a new Look/Compute phase

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Life cycles and asynchronicity δ

Look / Compute Move

/At each cycle, a robot is guaranteed to move by at least δ/

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Life cycles and asynchronicity

Look / Compute Move Look / Compute Move

/Different robots execute independent cycles, asynchronously/

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Pattern Formation problem: counterexample

x y x y x y x y x y x y

Let the initial configuration be rotationally symmetric

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Pattern Formation problem: counterexample

x y x y x y x y x y x y

All robots have the same view and compute symmetric destinations

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Pattern Formation problem: counterexample

x y x y x y x y x y x y

If they are all activated synchronously, they remain symmetric

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Pattern Formation problem: counterexample

x y x y x y x y x y x y

Hence Pattern Formation is unsolvable if the pattern is asymmetric

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Pattern Formation problem: state of the art

No pattern is formable from every possible initial configuration, except: Single point (aka Gathering problem) = ⇒ Solved [Cieliebak-Flocchini-Prencipe-Santoro, 2012]

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Pattern Formation problem: state of the art

No pattern is formable from every possible initial configuration, except: Single point (aka Gathering problem) = ⇒ Solved [Cieliebak-Flocchini-Prencipe-Santoro, 2012] Regular polygon = ⇒ Solved...

[Flocchini-Prencipe-Santoro-Viglietta, 2014–15]

...except for 4 robots! (aka Square Formation problem)

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Pattern Formation problem: state of the art

No pattern is formable from every possible initial configuration, except: Single point (aka Gathering problem) = ⇒ Solved [Cieliebak-Flocchini-Prencipe-Santoro, 2012] Regular polygon = ⇒ Solved...

[Flocchini-Prencipe-Santoro-Viglietta, 2014–15]

...except for 4 robots! (aka Square Formation problem)

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General approach to forming a regular polygon

An important configuration is the biangular one

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General approach to forming a regular polygon

An important configuration is the biangular one

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General approach to forming a regular polygon

The general algorithm identifies a supporting polygon...

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General approach to forming a regular polygon

g...And makes each robot move to the closest vertexg

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General approach to forming a regular polygon

As robots move, the supporting polygon is preserved

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General approach to forming a regular polygon

As robots move, the supporting polygon is preserved

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Why the general approach fails with 4 robots

With 4 robots, biangular configurations are rectangles

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Why the general approach fails with 4 robots

We can still identify a supporting square...

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Why the general approach fails with 4 robots

...But it is not unique!

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Why the general approach fails with 4 robots

...But it is not unique!

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Why the general approach fails with 4 robots

The “central” supporting polygon may be chosen...

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Why the general approach fails with 4 robots

...But asynchronous robots may never manage to form a square

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Why the general approach fails with 4 robots

...But asynchronous robots may never manage to form a square

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Why the general approach fails with 4 robots

...But asynchronous robots may never manage to form a square

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Why the general approach fails with 4 robots

...But asynchronous robots may never manage to form a square

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How to solve the rectangle

How do we solve the rectangular case?

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How to solve the rectangle

Choose a supporting square that is tilted by 45◦...

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How to solve the rectangle

...And make the robots move to the midpoints of its edges

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How to solve the rectangle

Again, the supporting square is preserved as the robots move

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How to solve the rectangle

Again, the supporting square is preserved as the robots move

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How to solve the rectangle

When they reach the midpoints, they form a square

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Identifying the supporting square

(In general, we can also identify a supporting square...(

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Identifying the supporting square

...Having a robot on each (extended) edge

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Identifying the supporting square

(But once again, the supporting square is not unique!(

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

Square Formation by Asynchronous Oblivious Robots

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

Square Formation by Asynchronous Oblivious Robots

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

Square Formation by Asynchronous Oblivious Robots

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

Square Formation by Asynchronous Oblivious Robots

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Identifying the supporting square

(However, there is a geometric construction that identifies one(

Square Formation by Asynchronous Oblivious Robots

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Identifying the supporting square

(All robots automatically agree on the same supporting square!(

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Identifying the supporting square

(All robots automatically agree on the same supporting square!(

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Identifying the supporting square

(All robots automatically agree on the same supporting square!(

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Identifying the supporting square

(No two robots have intersecting pathways!(

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Identifying the supporting square

(No two robots have intersecting pathways!(

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Identifying the supporting square

(No two robots have intersecting pathways!(

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Problem: orthogonal diagonals

Suppose the two diagonals “accidentally” become orthogonal

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Problem: orthogonal diagonals

Suppose the two diagonals “accidentally” become orthogonal

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Problem: orthogonal diagonals

gThen our construction does not workg

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Problem: orthogonal diagonals

The robots may not agree on a supporting square

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Special strategy for orthogonal diagonals

If the diagonals are orthogonal, we use a different approach

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Special strategy for orthogonal diagonals

If the diagonals are orthogonal, we use a different approach

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Special strategy for orthogonal diagonals

The robots that are closest to the center move away from it

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Special strategy for orthogonal diagonals

The robots that are closest to the center move away from it

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Special strategy for orthogonal diagonals

The robots that are closest to the center move away from it

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Special strategy for non-convex configurations

For non-convex configurations, our construction does not work...

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Special strategy for non-convex configurations

...Because the diagonals are not well defined

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Special strategy for non-convex configurations

gIn this case, the internal robot moves...g

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Special strategy for non-convex configurations

gIn this case, the internal robot moves...g

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Special strategy for non-convex configurations

...So to make the diagonals orthogonal...

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Special strategy for non-convex configurations

...And reduce the problem to the previous case

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Special strategy for collinear configurations

If the robots are collinear, the previous approach does not work

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Special strategy for collinear configurations

gIn this case, the internal robots move to either side of the lineg

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Special strategy for collinear configurations

As they asynchronously move, their supporting square may change

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Special strategy for collinear configurations

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So we must identify a “safe region”, e.g., a thin hexagon

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Special strategy for collinear configurations

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If the robots are in a thin hexagon, they follow a special algorithm

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Special strategy for collinear configurations

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If they end up on opposite sides of the long diagonal...

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Special strategy for collinear configurations

...We make them form a configuration with orthogonal diagonals

Square Formation by Asynchronous Oblivious Robots

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Special strategy for collinear configurations

Otherwise, they move on two vertices and wait for each other

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Special strategy for collinear configurations

Otherwise, they move on two vertices and wait for each other

Square Formation by Asynchronous Oblivious Robots

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Special strategy for collinear configurations

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Otherwise, they move on two vertices and wait for each other

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Special strategy for collinear configurations

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Otherwise, they move on two vertices and wait for each other

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Special strategy for collinear configurations

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Now that they are not moving, they agree on a supporting square

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General algorithm: one discordant robot

Suppose one robot is “discordant” with all the others

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General algorithm: one discordant robot

Suppose one robot is “discordant” with all the others

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General algorithm: one discordant robot

We let only the discordant robot move toward its final destination

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General algorithm: one discordant robot

As it moves, it may cause the diagonals to become orthogonal!

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General algorithm: one discordant robot

In this case, it has to stop at the point of orthogonality...

Square Formation by Asynchronous Oblivious Robots

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General algorithm: one discordant robot

In this case, it has to stop at the point of orthogonality...

Square Formation by Asynchronous Oblivious Robots

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General algorithm: one discordant robot

...So all robots will behave coherently, despite asynchronicity

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General algorithm: two opposite concordant, two finished

We let the two opposite robots move

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General algorithm: two opposite concordant, two finished

The diagonals can never become orthogonal by accident

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General algorithm: two opposite concordant, two finished

The diagonals can never become orthogonal by accident

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General algorithm: two opposite concordant, two finished

The diagonals can never become orthogonal by accident

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General algorithm: two opposite concordant, two finished

No thin hexagon can be formed by accident...

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General algorithm: two opposite concordant, two finished

No thin hexagon can be formed by accident...

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General algorithm: two opposite concordant, two finished

...Because the sum of distances from the long diagonal is too large

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General algorithm: two opposite concordant, two finished

...Because the sum of distances from the long diagonal is too large

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

...Because the sum of distances from the long diagonal is too large

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

...Because the sum of distances from the long diagonal is too large

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

...Because the sum of distances from the long diagonal is too large

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General algorithm: two opposite concordant, two finished

But the configuration may become non-convex by accident!

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General algorithm: two opposite concordant, two finished

But the configuration may become non-convex by accident!

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

But the configuration may become non-convex by accident!

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two opposite concordant, two finished

This can be prevented by making several shorter moves

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General algorithm: all concordant

(We let only the robots on the shortest diagonal move...(

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General algorithm: all concordant

(...Because it will remain the shortest as they move(

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General algorithm: all concordant

But one robot (not both!) may be “blocked” by the other diagonal

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General algorithm: all concordant

(If so, only the blocked robot moves, and stops on the diagonal(

Square Formation by Asynchronous Oblivious Robots

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General algorithm: all concordant

(Then all robots behave coherently as in the non-convex case(

Square Formation by Asynchronous Oblivious Robots

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General algorithm: two convergent robots

The convergent robots move, while the others wait

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General algorithm: two convergent robots

No thin hexagon can be formed by accident

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General algorithm: two convergent robots

No thin hexagon can be formed by accident

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General algorithm: two convergent robots

No thin hexagon can be formed by accident

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General algorithm: two convergent robots

No thin hexagon can be formed by accident

Square Formation by Asynchronous Oblivious Robots

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General algorithm: last case

If only one robot is external...

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General algorithm: last case

  • 25

>

...The angles it forms with the two far robots are > 25◦

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General algorithm: last case

So a thin hexagon cannot be formed, because its angles are 50◦

Square Formation by Asynchronous Oblivious Robots

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General algorithm: last case

So a thin hexagon cannot be formed, because its angles are 50◦

Square Formation by Asynchronous Oblivious Robots

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General algorithm: last case

This yields a simple coordination protocol for the robots in all cases

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Algorithm summary

The configuration is checked against each possible class, in the correct order!

1 Orthogonal diagonals 2 Thin hexagon 3 Non-convex 4 All concordant 5 Two convergent, two divergent 6 Two divergent, two divergent 7 One discordant Square Formation by Asynchronous Oblivious Robots

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Algorithm summary

The configuration is checked against each possible class, in the correct order!

1 Orthogonal diagonals 2 Thin hexagon 3 Non-convex 4 All concordant 5 Two convergent, two divergent 6 Two divergent, two divergent 7 One discordant

Ensure that, when a class transition occurs, No robot is moving (to prevent inconsistent behaviors!) The resulting class has lower index (in the list above)

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Algorithm summary

The configuration is checked against each possible class, in the correct order!

1 Orthogonal diagonals 2 Thin hexagon 3 Non-convex 4 All concordant 5 Two convergent, two divergent 6 Two divergent, two divergent 7 One discordant

Ensure that, when a class transition occurs, No robot is moving (to prevent inconsistent behaviors!) The resulting class has lower index (in the list above) The last rule is broken in only one case!

Square Formation by Asynchronous Oblivious Robots

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Resolving the anomaly

When all robots are on the same side of a thin hexagon...

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Resolving the anomaly

...They move to the vertices, and then apply the general algorithm

Square Formation by Asynchronous Oblivious Robots

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SLIDE 157

Resolving the anomaly

As a consequence, the internal robots move first

Square Formation by Asynchronous Oblivious Robots

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SLIDE 158

Resolving the anomaly

As a consequence, the internal robots move first

Square Formation by Asynchronous Oblivious Robots

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SLIDE 159

Resolving the anomaly

And finally the external robots move...

Square Formation by Asynchronous Oblivious Robots

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SLIDE 160

Resolving the anomaly

...Thus forming a square

Square Formation by Asynchronous Oblivious Robots

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SLIDE 161

Concluding remarks

The only solvable Pattern Formation problems for n robots are: Single point (except the case n = 2, which is unsolvable) Regular n-gon (now also for n = 4)

Square Formation by Asynchronous Oblivious Robots

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SLIDE 162

Concluding remarks

The only solvable Pattern Formation problems for n robots are: Single point (except the case n = 2, which is unsolvable) Regular n-gon (now also for n = 4) For n > 2, this is true even if Robots are fully synchronous Robots have a common notion of “clockwise” (chirality) Robots always reach their destination (rigidity) = ⇒ For Pattern Formation problems, these features are computationally irrelevant!

Square Formation by Asynchronous Oblivious Robots