Mobile Robots and distributed computing*

Mobile Robots and

distributed computing*

Maria Potop-Butucaru maria.gradinariu@lip6.fr

LIP6, UPMC Sorbonne Universités (Paris 6)

Real Robots

● UAVs(Unamed Aerial Vehiculs) & AUV (Autonomous underwater vehicles)

● Autonomous Ground Robots

● Metamorphic Robots

UAVs

●

Applications

− Military operations (Reconnaissance / Attack)

− Civilian applications (Firefighting/

Exploration)

●

Caracteristics

− Remote controlled

− Pre-programmed flight schedule

UAVs & AUVs

practical challenges

●

Distributed Cooperation

− UAVs Adaptive Flight schedule (in the air and for take off/landing)

− Pattern formation

− Connectivity (visual and/or communication)

●

Fault-tolerance

− Byzantine faults

− Crash faults

− Transient faults

Autonomous Ground Robots

●

Applications

− exclusive military operations

●

Caracteristics

− Sensing capabilities (LASAR – laser detection and ranging is currently used)

− Artificial inteligence embedded

●

Adaptive route planning

●

Adaptive motion (walk, trot,

run)

Autonomous Ground Robots practical challenges

●

Performant Sensors and cameras

●

Distributed Cooperation and coordination

●

Fault-tolerance

Metamorphic Robots

● Cooperative modules

− They can connect to each other and form complex structures

● Application

− only a lab toy for the

moment ...

Metamorphic Robots practical challenges

● Fully decentralization

● Fault tolerance

● Minimal

communication

Theoretical Models

Oblivious Planary Robots Suzuki and Yamashita '96

• Anonymous.

•

No direct communication.

•

No common coordinate system.

•

No common sens of direction (no common compas)

•

Memoryless.

No volume.

Execution Model

●

Fully Synchronous

●

SYm (semi-synchronous)

− Suzuki &Yamashita 96

●

CORDA (asynchronous)

−

Look

Compute

Move

SYm Model

Using its sensor/

camera, each

robot takes a snapshot of the world .

The visibility can be

 Limited

 Unlimited Look

t

SYm Model

Each robot executes an algorithm having as

input the position of the robots taken in the last Look phase

Look

t

Compute

SYm Model

t Look

Compute

Move

t+1

SYm Model

The robots move towards the

computed destination

t Look

Compute

Move

t+1

SYm Model

The robots move towards the

computed destination

t Look

Compute

Move

t+1

SYm Model

The robots move towards the

computed destination

t Look

Compute

Move

t+1

SYm Model

The robots move towards the

computed destination

Each robot executes this cycle

atomically

t Look

Compute

Move

t+1

CORDA Model

The first three steps are similar but

each step takes a

finite time different for each robot

Look

Compute

Move

CORDA Model

Look

Compute Move

Wait

CORDA Model

A wait step is added to take in account an

asynchronous behavior

Look

Compute Move

Wait

CORDA Model

A wait step is added to take in account an

asynchronous behavior

Look

Compute Move

Wait

CORDA Model

A wait step is added to take in account an

asynchronous behavior

Look

Compute Move

Wait

CORDA Model

A wait step is added to take in account an

asynchronous behavior

No global clock

Look

Compute Move

Wait

CORDA Model

A wait step is added to take in account an

asynchronous behavior

No global clock

No synchronization

Look

Compute Move

Wait

Model of the execution

●

Any problem solvable in CORDA model is also

solvable in the ATOM model

●

Any impossibility result for ATOM model holds for

CORDA model

Look

Compute

Move

Schedulers/Adversary

• centralized

• k-bounded

• regular

• synchonous

• asynchronous

• fair

Assumptions

 Full compass: Axes and polarities

of both axes. _x

y

x

y x

y

x

y

 Full compass: Axes and polarities of both axes.

 Half compass: Both axes known, but positive polarity on only one axis

x

x

x x

Assumptions

Assumptions

 Full compass: Axes and polarities of both axes.

 Half compass: Both axes known, but positive polarity on only one axis

 Direction only : Both axis but not

polarities

Assumptions

 Full compass: Axes and polarities of both axes.

 Half compass: Both axes known, but positive polarity on only one axis

 Direction only : Both axis but not polarities

 Polarity only: No common axis but common sense of left and right

x y

x

y x

y

x

y

Assumptions

 Full compass: Axes and polarities of both axes.

 Half compass: Both axes known, but positive polarity on only one axis

 Direction only : Both axis but not polarities

 Polarity only: No common axis but common sense of left and right

 No compass: ^{No common}

orientation

Cooperative Tasks

• Gathering

• Patern formation

• Election

• Flocking

The limits of Distributed

Computing

ANONYMOUS ROBOTS

●

Undistinguishable configuration - source of impossibility results

ANONYMOUS ROBOTS

●

Undistinguishable configuration - source of impossibility results

ANONYMOUS ROBOTS

●

Undistinguishable configuration - source of impossibility results

ANONYMOUS ROBOTS

●

Undistinguishable configuration - source of impossibility results

NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation

(x=x+1 / current_position=current_position

+1)

NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation

(x=x+1 / current_position=current_position

+1)

NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation

(x=x+1 / current_position=current_position

+1)

NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation

(x=x+1 / current_position=current_position

+1)

NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation

(x=x+1 / current_position=current_position

+1)

NON ATOMIC ACTIONS

x _i =a (atomic) ≠ Position _i =point_a (non atomic)

A robot can be stopped by the scheduler before it reaches its calculated destination.

Instruct the robots to move to the

internal concentric circle.

NON ATOMIC ACTIONS

x _i =a (atomic) ≠ Position _i =point_a (non atomic)

A robot can be stopped by the scheduler before it reaches its calculated destination.

Instruct the robots to move to the

internal concentric circle.

NON ATOMIC ACTIONS

x _i =a (atomic) ≠ Position _i =point_a (non atomic)

A robot can be stopped by the scheduler before it reaches its calculated destination.

Instruct the robots to move to the

internal concentric circle.

NON ATOMIC ACTIONS

x _i =a (atomic) ≠ Position _i =point_a (non atomic)

A robot can be stopped by the scheduler before it reaches its calculated destination.

Instruct the robots to move to the

internal concentric circle.

NON ATOMIC ACTIONS

x _i =a (atomic) ≠ Position _i =point_a (non atomic)

A robot can be stopped by the scheduler before it reaches its calculated destination.

Instruct the robots to move to the

internal concentric circle.

OBLIVIOUS ROBOTS (no past memory)

• Positive side:

• large scale

• inherently self-stabilizing

• Negative side:

• Robots cannot use the previous states information.

• Difficult to compose algorithms

ROBOTS AGREEMENT

The agreement invariant

• to share the same (an approximate) location

• gathering (convergence)

• to not share the same location

• ^scattering

• to acknowledge the same cheef

• leader election

• to form and maintain the same pattern

• flocking, pattern formation, connectivity

Agreement QoS

• Agreement time

• Faults resilience

• Scheduler freedom

The Gathering Problem

● Given a set of robots with arbitrary initial position and no initial agreement on a

global coordinate system they reach the exact same position in a finite number of

steps

The Gathering Problem

● Given a set of robots with arbitrary initial position and no initial agreement on a

global coordinate system they reach the exact same position in a finite number of

steps

Fault-free Gathering

● Suzuki&Yamashita proved that 2 gathering is deterministically impossible with oblivious robots

● 2-gathering probabilistically possible in ATOM model

− with probability p move to the location of another robot;

with probability (1-p) keep the current location

− Arbitrary scheduler, expected convergence time 2 rounds

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 1 ROBOT 2

Fault-free Gathering

● 2-gathering deterministically and probabilistically impossible in CORDA model under an arbitrary scheduler

ROBOT 2 ROBOT 1

Fault-free Gathering

● n-gathering is probabilistically possible under a bounded scheduler

− Choose with probability p=1/n a robot in the network and move toward this robot (with

probability 1-1/n the current position is not changed)

− Convergence time is n ² in expectation

Crash-tolerant gathering

● (n,1) gathering is deterministically impossible

under fair centralized and bounded schedulers

Crash-tolerant gathering

● (n,1) gathering is probabilistically impossible under a centralized scheduler

− Idea of the proof: a correct node can always travel between a faulty and another correct node

● (n,1) gathering is probabilistically possible under bounded schedulers

− Idea of the proof: with high probability all robots will join the faulty robot

● When f nodes are faulty, gathering is impossible so a weaker

version of the problem is considered (only correct nodes are

required to gather)

Byzantine-tolerant gathering

● (n,1)-gathering is possible under a centralized scheduler when:

− n>3, n odd and nodes have multiplicity knowledge

− n>1, n even and the scheduler is k<(n-1) bounded

Byzantine-tolerant gathering

● there is no deterministic algorithm that solves (n,f)-gathering (f>1) under a centralized k-

bounded scheduler with k≥[(n-f)/f] when n is

even and with k≥[(n-f)/(f-1)] when n is odd, even

when nodes are aware of the system multiplicity

Byzantine-tolerant gathering

● probabilistic (n,f)-gathering is possible under a bounded scheduler and multiplicity detection

− If member of a group with maximal multiplicity then probabilistically move to a group with the same multiplicity

− If not member of a group with maximal multiplicity

go to a group with maximal multiplicity

The convergence problem

For every ε > 0, there is a time t _ε after which all correct

robots are within distance of at most ε of each other.

The convergence problem

For every ε > 0, there is a time t _ε after which all correct

robots are within distance of at most ε of each other.

The convergence problem

For every ε > 0, there is a time t _ε after which all correct

robots are within distance of at most ε of each other.

Necessary and sufficient conditions

• Shrinking property (necessary)

• the diameter of correct positions decreases by a constant factor

0<α<1.

• Cautious property (sufficient)

• all calculated destinations are inside the range of correct positions -

similar to the approximate agreement

Byzantine-tolerant convergence

Byzantine-tolerant convergence

Byzantine-tolerant convergence

The scattering problem

– Closure - starting from a configuration where no two robots are located at the same position, no two robots are located at the same position thereafter.

– Convergence - regardless of the initial position of

the robots no two robots are eventually located at the

same position.

Impossibility of Deterministic Scaterring

● Intuition : two robots on the same point may

choose exactly the same destination point

Impossibility of Deterministic Scaterring

● Intuition : two robots on the same point may

choose exactly the same destination point

Impossibility of Deterministic Scaterring

● Intuition : two robots on the same point may

choose exactly the same destination point

Probabilistic Scaterring*

● Intuition : two robots chose probabilisticaly a

destination point

Probabilistic Scaterring*

● Intuition : two robots chose probabilisticaly a

destination point

Probabilistic Scaterring*

● Intuition : two robots chose probabilisticaly a

destination point

Probabilistic Scaterring*

● Intuition : two robots chose probabilisticaly a

destination point

Probabilistic scattering*

Probabilistic scattering*

r1

Probabilistic scattering*

r ₂

r1

Probabilistic scattering*

r ₂

r1

Probabilistic scattering*

r ₂

r1

Probabilistic scattering*

r ₂

r1

Probabilistic scattering*

r ₂

r1

Probabilistic scattering*

r ₂

r1

Scattering with Limited Visibility*

8

● Each robot can see the locations of robots within (fixed) visible range

− Range is unit distance (common among all robots)

Visibility Graph

9

● Graph representing visibility relationship

Connectivity-Preserving Property

10

● No cooperation is possible under the disconnection of visibility graph

− The system behave as two independent groups of

robots

Connectivity-Preserving Property

10

● No cooperation is possible under the disconnection of visibility graph

− The system behave as two independent groups of

robots

Connectivity-Preserving Property

10

● No cooperation is possible under the disconnection of visibility graph

− The system behave as two independent groups of robots

Task must be accomplished with preserving

the connectivity of visibility graphs

Risk of Disconnection(1/2)

12

● Robots on the border of visibility range

● To scatter P, robot on that point must move via randomization

multiple robots

P

Q R

Risk of Disconnection(2/2)

13

● Any movement yields direct disconnection to Q or R

P

Q R P

Q R

Risk of Disconnection(2/2)

13

● Any movement yields direct disconnection to Q or R

P

Q R P

Q R

Risk of Disconnection(2/2)

14

● Any movement yields direct disconnection to Q or R

− With non-zero probability, all robots on P will move

P

Q R P

Q R

Risk of Disconnection(2/2)

14

● Any movement yields direct disconnection to Q or R

− With non-zero probability, all robots on P will move

P

Q R P

Q R

Lower Bound

15

● The following configuration gives Ω (n)-lower bound

・・・ ・・・

2 robots

(n/2) - 1 robots (n/2) - 1 robots

P

Blocked Location

16

● Not all robot-on-boundary situations take the risk

P Q

R S

Q

R

S

P

Blocked Location

16

● Not all robot-on-boundary situations take the risk

P Q

R S

Q

R

S

Blocked Location (Definition)

17

● A location P is blocked

No arc of center angle less than π can contain all robots on the visibility boundary

P

P R

S

R

Blocked Non-blocked

Blocked Location (Definition)

17

● A location P is blocked

No arc of center angle less than π can contain all robots on the visibility boundary

P

P R

S

R

Blocked Non-blocked

O(n) Scattering algorithm

18

● Any non-blocked robot with boundary robots moves

− the direction bisecting covering arc

− half of the distance to the chord of covering arc

(Note: Even the robot on single point must move to make other robot non-blocked)

P

R

If P is multiple we further divide the travel to create two possible

destinations (probabilistically chosen)

P

Proving Connectivity

19

● The case that looks like problem

− Concurrent movement of two robots that sees each other on their boundaries

● The destinations are necessarily contained in a disc of unit diameter

P

R

Proving Connectivity

19

● The case that looks like problem

− Concurrent movement of two robots that sees each other on their boundaries

● The destinations are necessarily contained in a disc of unit diameter

P

R

Proving Connectivity

19

● The case that looks like problem

− Concurrent movement of two robots that sees each other on their boundaries

● The destinations are necessarily contained in a disc of unit diameter

P

R

Proving Connectivity

19

● The case that looks like problem

− Concurrent movement of two robots that sees each other on their boundaries

● The destinations are necessarily contained in a disc of unit diameter

P

R

Time Complexity

● Always at least one robot is non-blocked

− The corners of convex hull

20

Blocked Non-blocked

Time Complexity

● Always at least one robot is non-blocked

− The corners of convex hull

20

Blocked Non-blocked

By the movement of all non-blocked robots,

at least one blocked robot becomes non-blocked

All robots becomes non-blocked within O(n) rounds

Scattering completes within O(log n) expected rounds +

Diameter-Sensitive Bounds

21

● Counting the number of “shells” at initial configuration

Cercle Formation problem problem*

• Robots eventually form a cercle

Cercle Formation problem problem*

• Robots eventually form a cercle

Cercle Formation problem problem*

• Robots eventually form a cercle

Cercle Formation problem problem*

• Robots eventually form a cercle

Cercle Formation problem problem*

• Robots eventually form a cercle

Cercle Formation problem problem*

• Robots eventually form a cercle

The Leader Election problem*

• Given a set of robots they

eventually agree on an unique

leader

Impossibility of deterministic leader

election

Trivial 3 robots election

If my angle is different (the smallest or the greatest) then I am leader.

α

α

ß ß

• If perfect symmetry then with some

probability p change the curent position

ß

ß

α

Trivial 3 robots election

Leader Election for n robots

The Flocking Problem*

• Leader election

• Agreement on the Flocking formation

• Motion

Move to the SEC

Circular Formation

p₁ p₂

p₃ p₄

p₅

p₆

r

r

r

r

r

r

r

p₇

p₁ p₃

p₄

p₅

p₆

r

r

r

r

r

r

p₇

r

r

r

r

r

r

r

Circular Formation

p₁ p₂

p₃ p₄

p₅

p₆

r

r

r

r

r

r

r

p₇

p₁ p₃

p₄

p₅

p₆

r

r

r

r

r

r

p₇

r

r

r

r

r

r

r

Formation de flocking

Motion Formation

y

x

∀ i, ∃ j ^ x ⁱ = -x ^j ∀ i ^≠ j

Motion Formation

Reference

y

x

∀ i, ∃ j ^ x ⁱ = -x ^j ∀ i ^≠ j

Motion Formation

Reference

Head

y

x

∀ i, ∃ j ^ x ⁱ = -x ^j ∀ i ^≠ j

Global Algorithm

p₃ p₄

p₅

p₆

r

r

r

r

p₇

r

r

r

r

r

r

r

Open problems

• Model Transformation

–Lamport registers and ATOM&CORDA

–ATOM, CORDA and synchronous/asynchronous

• New Models

–What is between ATOM and CORDA ? –CORDA + k-bounded ??? ATOM

–Additional assumption (e.g. volumic robots, memory etc)

–Computational power of the model

• Tasks