System Models & Basic Terminology - Hiroshima University...Xavier DÉFAGO School of Information...

Xavier DÉFAGOSchool of Information Science

Japan Advanced Institute of Science and Technology

Hiroshima, December 2009

Cooperative Mobile RobotsSystem Models & Basic Terminology

PDCAT Tutorial

X. Défago. “Tutorial: cooperative mobile robots: system models & basic terminology”. Hiroshima, Japan, December 2009.

Simple Illustration

Context & MotivationBottom-Up

1./ Build robots2./ Enforce coordination=> Realistic=> Coordination is an afterthought

Emergence / Stigmergy1./ Define simple behavior2./ Study combined behavior3./ Try to infer rules=> Works well ... on average cases

Context & MotivationAlgorithmic Approach

1./ Define model & problem2./ Find minimal set of assumptions3./ Prove possibility / impossibility=> Fundamental limits of coordination

HighlightsTop-down approachAims for self-organizing propertiesCombines:• Distributed algorithms, computational geometry

OutlineSystem Models

Basic Model

Model VariantsSynchrony, etc...

Main ProblemsDefinitionsSome Known Results

Leads for Discussion

System Models

System OverviewEnvironment

2D Euclidean planeNo landmarks / obstacles

RobotsProcess + LocationPrivate coordinate systemLOOK - COMPUTE - MOVE

InteractionsObservations, no comm.Activations

Basic Model

Synchrony

Environment

Sensing

Knowledge

Interactions

ModelOriginal Model

Called: SYm, SSYNC, ATOM.Starting point for other models/variants

[SY99] Suzuki, Yamashita. Distributed Anonymous Mobile Robots: formation of geometric patterns. SIAM J. Comp., 28(4):1347–1363 (1999).

SSYNC Atom

Environment2D Euclidean Space

Continuous coordinatesEuclidean distanceNo boundariesNo landmarksNo obstacles

# similar w.r.t. symmetry

Z = (o, d, u)

EnvironmentCoordinate system

Origin, direction of x-axis, unit distanceGlobal system is unknown to robots

TimeDiscrete time: 0,1,2,...Time is unknown to robots

Z = (O, !x, !y)

RobotsRobots in the System

Robots are dimensionless (i.e., points)Can movePossibly occupy the same locationAre undistinguishableExecute the same codeDo not know their identifier (anonymous)• Disallows:

if me == then ...

{r1, r2, . . . , rn}

RobotsLocal Coordinates

Common sense of orientation (chirality):• x,y-axes at 90º counter-clockwise

Let’s look at robot

Zi = (!oi, di, ui)

RobotsPosition

: position of robot at time

Initial Position

Positions Multiset => several robots @ same positionPosition distinct initially,

riri tpi(t)

P (t) = {pi(t)|1 ! i ! n}

!i, j : i "= j # pi(0) "= pj(0)

RobotsAlgorithm

Sequential processRobot is: active | inactiveActivated randomly

ActivationLOOK at the environmentCOMPUTE a target destinationMOVE toward the target

LOOKNotation in terms of

ObservationGet a set of points

Get its own position

Zi[p]i ! p

[P (t)]i = {[pk(t)]i |1 ! k ! n}

symmetrical

COMPUTECompute Target Destination

Outputs a single point

Function is deterministicComputes target destination

Variants:Oblivious: function is statelessNon-oblivious: retains past observations

!([P (t)]i , [pi]i) !" [p!i(t)]i

Robotri

MOVEMove Toward Target

Move toward target destinationLimited movement• Robot has reachability Δi > 0 (unit distance)• Moves to point within Δi nearest to target• Δi > 0 is not infinitesimally small

New position reached before time

Activation ScheduleActivation

Infinite sequence: : set of active robots at

ConditionsAt each time: at least one robot is activeFor every robot: it is active infinitely-many times

Active RobotExecutes: LOOK - COMPUTE - MOVE

A = A0, A1, . . . , At, . . .At t

Activation Schedule

compute

Special CasesSynchronized:Centralized:

!t : "At" = n!t : "At" = 1

Gathering Problem

InitiallyRobots located arbitrarily

ObjectiveEventually all robots at same location

FormationMust reach goal in finite number of steps

Difficulty of Gathering

IllustrationTwo robots A and BSYm modelDeterministic algorithmOblivious robots=> 2-Gathering impossible [SY99]

Model Variants

Activation ScheduleOriginal Model

Called: SYm, SSYNC, ATOM

Asynchronous

Called: CORDA, ASYNCSeparate events:• LOOK• COMPUTE• BEGIN MOVE• END MOVE

[FPSW05] Flocchini, Prencipe, Santoro, Widmayer. Gathering of asynchronous robots with limited visibility. T.C.S., 337:147–168, 2005.

CORDA Model

Important PointsRobots can be seen while movingDifferent robot “speed”

Special CasesSYm strictly included in CORDA

look compute move

MultiplicityNo multiplicity detection

Set of pointsTwo+ collocated robots seen as single point.

Multiplicity detectionPoint of multiplicity identified (weak)Count of collocated robots (strong)

VisibilityFull visibility

All robots included in the set P(t)

Limited visibilityVisibility range RRobot ri gets subset:• Only the robots within distance R

Usually:• Initially, visibility graph is connected

MemoryNon-oblivious

Input includes all / part of past activationsActivation may include stateful information

ObliviousInput is last observation onlyActivation is stateless

IdentityAnonymous

Robots have no identity information

With IdentitySome robot can be designated as “leader”Allows: if me == “leader” then ...NB: Identity is visible or not

DeterminismDeterministic

Algorithm allows only deterministic operations

RandomizedAlgorithm allows probabilistic operations

VolumeDimensionless Robots

Robots can share the same locationRobots have no size

Robots with VolumeRobots have a sizeRobots should not collide

MovementInstantaneous

Robot “teleports” to the target

Duration / PathMovement takes some timeRobots follows a pathAdditional possibility of collision

NoteNearly identical for SYm model

CompassNo assumption

Original SYm model

CompassRobots agree on a common directionNo agreement on origin

Eventual CompassRobots eventually agree on a common dir.

Faulty CompassesRobots agree within some angle errorStatic / dynamic variants

SchedulersFairness

Unfair: may active some robots unfairlyFair: all robots activated infinitely-often

k-BoundedAt most k activations of any robot r’between 2 activations of some robot r

CentralizedAt most 1 robot active

Fault-ToleranceCrash

Some robots may crash:• Disappear from the system (trivial)• Stop moving

ByzantineSome robots move against the algorithmStrong/weaker than schedulerAware/unaware of scheduler decisionAware/unaware of algorithm decisions

Error-ToleranceErrors

Errors on observationsUsually:• Errors are relative wr.t. robot’s location• Errors are bounded• Bounds are known

Problems

Gathering / ScatteringGathering

Initially: arbitrary configuration=> all robots share same location• permanently• location is not predetermined

ScatteringInitially: robots possibly at same location=> all robots have distinct positions• permanently

NB: Dual problems 37

Pattern FormationGeneral Pattern

Initially: arbitrary configurationGiven: geometric shape=> Organize into the shape

Special CasesCircle formation (regular n-gon)Point formation (gathering)

RelatedLeader election

FlockingFlocking

Initially: arbitrary configuration=> Robots eventually move together

VariantsMove while keeping given shapeMove while following given pathAssumes leader or not...

DiscussionsSummary

SYm model, CORDA modelMany model variantsMany fundamental problems

ApproachFormal modelProofs of correctness

AimMinimum requirements for cooperation

Important References•[SY99] Suzuki, Yamashita. Distributed anonymous mobile robots:

formation of geometric patterns. SIAM J. Comp., 28(4):1347–1363 (1999).

•[GP04] Gervasi, Prencipe: Coordination without communication: the case of the flocking problem. Discr. Applied Math. 144(3): 324-344 (2004)

•[FPSW05] Flocchini, Prencipe, Santoro, Widmayer: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci. 337(1-3): 147-168 (2005)

•[Pre05] Prencipe. The Effect of Synchronicity on the behavior of autonomous mobile robots. Theory Comput. Syst. 38(5): 539-558 (2005).

•[AP06] Agmon, Peleg. Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM J. Comput. 36(1): 56-82 (2006)

•[DGMR06] Défago, Gradinariu Potop-Butucaru, Messika, Raipin Parvédy: Fault-Tolerant and Self-stabilizing Mobile Robots Gathering. DISC 2006: 46-60

•[IKIW07] Izumi, Katayama, Inuzuka, Wada: Gathering autonomous mobile robots with dynamic compasses: optimal result. DISC 2007: 298-312

•[Pre07] Prencipe. Impossibility of gathering by a set of autonomous mobile robots. Theor. Comput. Sci. 384(2-3): 222-231 (2007)

•[DS08] Défago, Souissi. Non-uniform circle formation algorithm for oblivious mobile robots with convergence toward uniformity. Theor. Comput. Sci. 396(1-3): 97-112 (2008)

•[FPSW08] Flocchini, Prencipe, Santoro, Widmayer. Arbitrary pattern formation by asynchronous, anonymous, oblivious robots. Theor. Comput. Sci. 407(1-3): 412-447 (2008)

•[CP08] Cohen, Peleg. Convergence of autonomous mobile robots with inaccurate sensors and movements. SIAM J. Comp. 38(1): 276-302 (2008)

•[DP08] Dieudonné, Labbani-Igbida, Petit. Circle formation of weak mobile robots. ACM Trans. Autonomous Adapt. Syst., 3(4): (2008)

•[BGT09] Bouzid, Gradinariu Potop-Butucaru, Tixeuil. Optimal Byzantine resilient convergence in asynchronous robots networks. SSS 2009: 165-179

•[DP09] Dieudonné, Petit. Scatter of robots. Parallel Processing Letters 19(1): 175-184 (2009)

•[SDY09] Souissi, Défago, Yamashita. Using eventually consistent compasses to gather memory-less mobile robots with limited visibility. ACM Trans. Autonomous Adapt. Syst., 4(1): (2009)

•[SIW09] Souissi, Izumi, Wada: Oracle-based flocking of mobile robots in crash-recovery model. SSS 2009: 683-697

System Models & Basic Terminology - Hiroshima University...Xavier DÉFAGO School of Information...

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