Mobile Robots and distributed computing*tixeuil/m2r/uploads/Main/robots-cours1.pdf · Crash-tolerant gathering (n,1) gathering is probabilistically impossible under a centralized

Mobile Robots and distributed computing*

Maria [email protected]

LIP6, UPMC Sorbonne Universités (Paris 6)

lundi 11 février 2013

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

RQ-2 Pioneer used in the Golf war


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)

DARPA – BIG DOG


Autonomous Ground Robotspractical challenges

● Performant Sensors and cameras

● Distributed Cooperation and coordination

● Fault-tolerance

DARPA – BIG DOG


Metamorphic Robots

● Cooperative modules−They can connect to

each other and form complex structures

● Application− only a lab toy for the

moment ...

S-bot project


Metamorphic Robots practical challenges

● Fully decentralization● Fault tolerance● Minimal

communication

S-bot project


Theoretical Models

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Oblivious Planary RobotsSuzuki 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 ModelUsing its sensor/

camera, each robot takes a snapshotof the world .

The visibility can be Limited Unlimited

Lookt


SYm Model

Each robot executes an algorithm having as input the position of the robots taken in the last Look phase

Lookt

Compute


SYm Model

Lookt

Compute

Movet+1


SYm ModelThe robots move

towards the computed destination

Lookt

Compute

Movet+1




Lookt

Compute

Movet+1




Lookt

Compute

Movet+1




Each robot executes this cycle atomically

Lookt

Compute

Movet+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 ModelA wait step is added to

take in account an asynchronous behavior

Look

Compute

Move

Wait




Look

Compute

Move

Wait




Look

Compute

Move

Wait




No global clock

Look

Compute

Move

Wait




No global clockNo 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

22


AssumptionsFull 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

xx

Assumptions



of both axes.Half compass: Both axes known,

but positive polarity on only one axisDirection only: Both axis but not

polarities





polarities Polarity only: No common axis but

common sense of left and right

x

y

x

y

x

y

x

y





polarities Polarity only: No common axis but

common sense of left and rightNo compass: No common

orientation


Cooperative Tasks• Gathering

• Patern formation

• Election

• Flocking


The limits of Distributed Computing

29


ANONYMOUS ROBOTS

● Undistinguishable configuration - source of impossibility results


ANONYMOUS ROBOTS



ANONYMOUS ROBOTS



ANONYMOUS ROBOTS



NO COMMON COORDINATE

Functions with different semantic:

Maximum (among a set of positions)

Incrementation (x=x+1 / current_position=current_position

+1)






+1)






+1)






+1)






+1)


NON ATOMIC ACTIONS

xi=a (atomic) ≠ Positioni=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





NON ATOMIC ACTIONS





NON ATOMIC ACTIONS





NON ATOMIC ACTIONS





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

34


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

37


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

37


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




ROBOT 1 ROBOT 2




ROBOT 1 ROBOT 2




ROBOT 1 ROBOT 2




ROBOT 1 ROBOT 2




ROBOT 1 ROBOT 2




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 n2 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



● 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



● 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 problemFor every ε > 0, there is a time tε 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






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.

61lundi 11 février 2013

Impossibility of Deterministic Scaterring

● Intuition : two robots on the same point may choose exactly the same destination point

56




56




56


Probabilistic Scaterring*● Intuition : two robots chose probabilisticaly a

destination point

57

Dieudonne et al. 2009



destination point

57




destination point

57




destination point

57



Probabilistic scattering*

62

Malekova, Potop-Butucaru, Tixeuil 2009



62

r1




r2

62

r1




r2

62

r1




r2

62

r1




r2

62

r1




r2

62

r1




r2

62

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)

Izumi, Potop-Butucaru, Tixeuil 2010


Visibility Graph9

● 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




robots




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

PQ R



● Any movement yields direct disconnection to Q or R

PQ R PQ R




PQ R PQ R




− With non-zero probability, all robots on P will move the same position →　disconnection

PQ R PQ R




− With non-zero probability, all robots on P will move the same position →　disconnection

PQ R PQ R


Lower Bound15

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

・・・・・・

2 robots

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


P

Blocked Location16

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

P

Q

R

S

Dangerous: Any movement by Pcauses a disconnection

Q

R

SSafe: P can move along the red arrow


P

Blocked Location16

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

P

Q

R

S

Dangerous: Any movement by Pcauses a disconnection

Q

R

SSafe: P can move along the red arrow


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

Q

R

S

Q

R

S

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

Q

R

S

Q

R

S

Blocked Non-blocked

covering arc


O(n) Scattering algorithm18

● 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

Q

R

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

destinations (probabilistically chosen)

Pdestination 1

destination 2


Proving Connectivity19

● 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

PR






PR






PR






PR


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 Bounds21

● Counting the number of “shells” at initial configuration

Upper bound : O(D2)(Probably, not tight)


Cercle Formation problem problem*

• Robots eventually form a cercle

Xavier Défago et. al. 2002






















The Leader Election problem*

• Given a set of robots they eventually agree on an unique leader

Canepa & Potop-Butucaru 2007


Impossibility of deterministic leader election


Trivial 3 robots election If my angle is different (the smallest or the greatest) then I

am leader.

α

αß 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

Canepa & Potop-Butucaru 2007


Move to the SEC


Circular Formation

p1p2

p3

p4

p5

p6

r1

r2

r3

r4

r5r6

r7

p7

p1p2

p3

p4

p5

p6

r1

r2

r3

r4

r5r6

r7

p7

r1

r6r7

r3

r5

r4

r2


Circular Formation

p1p2

p3

p4

p5

p6

r1

r2

r3

r4

r5r6

r7

p7

p1p2

p3

p4

p5

p6

r1

r2

r3

r4

r5r6

r7

p7

r1

r6r7

r3

r5

r4

r2

Formation de flocking


Motion Formationy

x

∀i, ∃j xi = -xj ∀i≠j

if |xi | > |xj | than |yi | < |yj |


Motion Formation

Reference

y

x




Motion Formation

Reference

Head

y

x




Global Algorithm

p1p2

p3

p4

p5

p6

r2

r3

r4

r5r6

p7

r1

r6r7

r3

r5

r4

r2


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–lower/upper bounds

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Mobile Robots and distributed computing*tixeuil/m2r/uploads/Main/robots-cours1.pdf · Crash-tolerant gathering (n,1) gathering is probabilistically impossible under a centralized