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

2

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

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

3

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

Context & MotivationAlgorithmic Approach

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

4

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

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

OutlineSystem Models

Basic Model

Model VariantsSynchrony, etc...

Main ProblemsDefinitionsSome Known Results

Leads for Discussion

5

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

System Models

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

System OverviewEnvironment

2D Euclidean planeNo landmarks / obstacles

RobotsProcess + LocationPrivate coordinate systemLOOK - COMPUTE - MOVE

InteractionsObservations, no comm.Activations

7

Basic Model

Synchrony

Environment

Sensing

Knowledge

Interactions

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

ModelOriginal Model

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

8

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

SSYNC Atom

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

Environment2D Euclidean Space

Continuous coordinatesEuclidean distanceNo boundariesNo landmarksNo obstacles

9

!"

# similar w.r.t. symmetry

Z = (o, d, u)

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

EnvironmentCoordinate system

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

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

10

Z = (O, !x, !y)

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

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 ...

11

{r1, r2, . . . , rn}

rx

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

RobotsLocal Coordinates

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

12

Let’s look at robot

ri

!oi

!xi

di

ui

!yi

O !x

Zi = (!oi, di, ui)

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

RobotsPosition

: position of robot at time

Initial Position 

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

13

Let’s look at robot

riri tpi(t)

pi(0)

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

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

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

RobotsAlgorithm

Sequential processRobot is: active | inactiveActivated randomly

ActivationLOOK at the environmentCOMPUTE a target destinationMOVE toward the target

14

Let’s look at robot

ri

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

LOOKNotation  in terms of

ObservationGet a set of points

Get its own position

15

Zi[p]i ! p

Let’s look at robot

ri

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

r1

r3r4

r2x

x

xx

symmetrical

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

COMPUTECompute Target Destination

Outputs a single point

Function is deterministicComputes target destination

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

16

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

Robotri

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

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

17

t + 1

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

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

18

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

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

Activation Schedule

19

rc

rb

ra

look

compute

move

t

At

t + 1

Special CasesSynchronized:Centralized:

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

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

Gathering Problem

InitiallyRobots located arbitrarily

ObjectiveEventually all robots at same location

FormationMust reach goal in finite number of steps

20

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

Difficulty of Gathering

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

21

A B

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

Model Variants

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

Activation ScheduleOriginal Model

Called: SYm, SSYNC, ATOM

Asynchronous

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

23

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

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

CORDA Model

Important PointsRobots can be seen while movingDifferent robot “speed”

Special CasesSYm strictly included in CORDA

24

rc

rb

ra

look compute move

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

MultiplicityNo multiplicity detection

Set of pointsTwo+ collocated robots seen as single point.

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

25

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

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

26

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

MemoryNon-oblivious

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

ObliviousInput is last observation onlyActivation is stateless

27

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

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

28

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

DeterminismDeterministic

Algorithm allows only deterministic operations

RandomizedAlgorithm allows probabilistic operations

29

A B

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

VolumeDimensionless Robots

Robots can share the same locationRobots have no size

Robots with VolumeRobots have a sizeRobots should not collide

30

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

MovementInstantaneous

Robot “teleports” to the target

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

NoteNearly identical for SYm model

31

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

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

32

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

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

33

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

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

34

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

Error-ToleranceErrors

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

35

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

Problems

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

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

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

Pattern FormationGeneral Pattern

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

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

RelatedLeader election

38

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

FlockingFlocking

Initially: arbitrary configuration=> Robots eventually move together

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

39

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

DiscussionsSummary

SYm model, CORDA modelMany model variantsMany fundamental problems

ApproachFormal modelProofs of correctness

AimMinimum requirements for cooperation

40

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

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

41

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

•[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

42

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

•[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

43