Distributed Algorithms for Swarm Robotsigwoa/IGWOA_KM.pdf · Distributed Algorithms for Swarm Robots Krishnendu Mukopad-hyaya Introduction Computational model Examples of some problems

DistributedAlgorithmsfor Swarm

Robots

KrishnenduMukopad-

hyaya

Introduction

Computationalmodel

Examples ofsomeproblems

Arbitrarypatternformation

Leader election

Circle formation

Gathering

Fault tolerantgathering ofpoint robots

Gatheringunder unequalvisibility range

Conclusion

Distributed Algorithms for Swarm Robots

Krishnendu Mukopadhyaya

ACM UnitIndian Statistical Institute, Kolkata

Indo-German Workshop on Algorithms9-13 Feb 2015

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Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Outline

1 Introduction

2 Computational model

3 Examples of some problemsArbitrary pattern formationLeader electionCircle formationGathering

Fault tolerant gathering of point robotsGathering under unequal visibility range

4 Conclusion

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Leader election

Circle formation

Gathering



Conclusion

Swarm Robots

• Group of small, inexpensive, identical, autonomous, mobilerobots.

• Collaboratively executing work• moving large object, cleaning big surface.

• Geometric point of view: points moving on the 2Dplane.

• Tasks: Forming geometric patterns like point, circle etc.• Distributed in nature.

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Leader election

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Conclusion

General characteristics of swarm robots

• Point/Unit Disc

• Autonomous

• Identical

• No message passing

• Sense surroundings

• Move on the 2D plane

• Limited computational power

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Leader election

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Conclusion

Computational model

• Execute wait-look-compute-move cycle.

In wait state robots do nothing.

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Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Computational model


Look

Rv

r

• Rv (visibility range) can be limited or unlimited

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Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Computational model


t

Computer

• r computes its destination t

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Leader election

Circle formation

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Conclusion

Computational model


t

Mover

• r moves to t• SYm: Rigid motion.• CORDA: Non-rigid motion.

Obliviousness

After executing a cycle the robots forget all data.

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Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Computational model

• Execute wait-look-compute-move cycle synchronously.

r1

r2r3

r4

r5

• All robots look at the same time

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hyaya

Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Computational model

• Execute wait-look-compute-move cycle asynchronously.

r1

r3

r5

T1

T3

T5

r2

r4r6 r7

r8

• Different robots look, compute and move at differenttimes.

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Introduction

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Leader election

Circle formation

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Conclusion

Computational model

• Execute wait-look-compute-move cyclesemi-synchronously.

• An arbitrary set of robots look at the same time.

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Leader election

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Conclusion

Computational model

• Agreement on co-ordinate system.

X

Y

X

Y

X

Y

X

Y

X

Yr1

r2

r3

r4

r5

• Robots having same Sense of Directions (SoD) and samechirality

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Leader election

Circle formation

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Conclusion

Computational model


X

Y

X

Y

X

Y

X

Y

X

Y

r1

r2

r3

r4r5

• Robots having same SoD but different chirality

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Leader election

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Conclusion

Computational model


X

Y

X

Y

X

Y

XY

X

Y

r1

r2

r3

r4

r5

• Robots having different SoD but same chirality

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Conclusion

Computational model


XY

r2

XY

r4X

Y

r1

Y

r5

X

X

Y

r3

• Robots having different SoD and different chirality

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Introduction

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Leader election

Circle formation

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Conclusion

Examples of some problems

Goal

Coordination: formation of pattern for executing some job,

• moving an object.

• covering/painting an area

• guarding a geographical area etc.

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Circle formation

Gathering



Conclusion


• Arbitrary pattern formation [FPS2008]

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Leader election

Circle formation

Gathering



Conclusion

Arbitrary pattern formation

Solution approach

• A set of robots is selected for movement.

• While these robots are in motion this set of eligible robotsremains invariant.

• The set changes only after all the robots reach theirdestinations.

• In case of nonrigid motion, if a robot stops before reachingits destination, it is again selected for movement in itsnext cycle.

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Conclusion

Geometric characterization for pattern formation

Symmetric PatternsLine of Symmetry

Line of Symmetry

(a) (b) (c)

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Conclusion


Asymmetric Patterns

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Leader election

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Conclusion


Orderable set

• A set of points is called an orderable set, if there exists adeterministic algorithm, which produces a unique orderingof the points of the set, such that the ordering is sameirrespective of the choice of origin and coordinate system.

Theorem[GM2010]

• A symmetric pattern is not orderable.

• An asymmetric pattern is orderable.

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Results on pattern formation

Scheduling SoD Chirality ResultsAsync Yes No (i) Any pattern formable

with odd no. of robots(ii)Symmetric pattern is formable

for even no. of robots [FPSW1999].Async Yes Yes Arbitrary pattern is formable for

any no. of robots [FPSW2001].ASync No Yes Characterization of all patterns

formable from anyinitial configurations[FPS2008, YS2010].

Theorem [GM2010]

If a set of robots is orderable, then any asymmetric pattern canbe formed by them even without having common chirality andavoiding collisions.

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Leader election

Circle formation

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Conclusion


• Leader Election [DPV2010]

r1

r2

r3 r4

r5

L

• The robots elect r1 as their leader.

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Conclusion

Results on leader election

SoD Chirality # of robots ResultsYes Yes Any Leader election possible [FPSW1999].Yes No Odd Leader election possible [FPSW2001].No Yes/No Any Leader election not possible [FPSW2001].No Yes Any characterization of all

geometric positions [DP2007].No No Odd characterization of all

geometric positions [DP2007].No No Any Characterization of all

geometric positions whereiterative leader election

(total ordering of robots) is possible.[GM2010]

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Leader election

Circle formation

Gathering



Conclusion

Results on leader election

Theorem [DP2010]

Leader election and pattern formation problems are equivalent.

Theorem [GM2010]

A set of robots is orderable if and only if leader election ispossible.

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Leader election

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Gathering



Conclusion

Circle formation

Results on circle formation

Scheduling Visibility Agreement Resultsrange in co-ordinate

Sync Limited No Heuristic ofapproximate circle

formation[SS1990].

Ssync Unlimited No Circle formation[DK2002].

ASync Unlimited No Bi-angularCircle formation

[K2005].ASync Unlimited No Circle formation

[DS2008].Async Unlimited Agree in chirality Uniform Circle formation

by n 6= 4 robots[FGSV2014].

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Introduction

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Leader election

Circle formation

Gathering



Conclusion

Circle formation

Recent extension for fat robots [DDGM2012]

• Circle formation is possible for (i) transparent fat robots,(ii) with limited visibility, (iii) with agreement in SoD andChirality and (iv) without collision.

O X

Y

R

O X

Y

R Rv

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Conclusion

Circle formation

Recent extension for transparent fat robots [DDGM2013]

• Circle formation is possible for (i) transparent fat robots,(ii) with unlimited visibility ,(iii) without agreement inSoD and Chirality and (iv) without collision.

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Conclusion


Gathering (point robots)

• Gathering [P2007] or Convergence [CP2006]

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Conclusion


Gathering (fat robots)

• Gathering Fat Robots [CGP2009].

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Conclusion

Gathering

Solution approach

Find an point to gather, such that the point will not change till the robots gather.

• Centroid or center of gravity (CoG): Given a set of n points represented bytheir coordinate values as {(x1, y1), . . . , (xn, yn)}. COG of the points is

given by (xc, yc) where xc = x1+...+xnn

and yc = y1+...+ynn

.

• Weber point: Given a set of n points, Weber point is the point whichminimizes the sum of distances between itself and all the points. This isalso known as the Fermat or Torricelli point. Weber point does not changeif the points are moving towards it. Not computable for ≥ 5 points.

• Center of Minimum Enclosing Circle (MEC): It is an invariant point if thepoints defining the MEC do not move and no robot moves outside theMEC.

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Conclusion

Gathering

Difficulties

• Asynchronous: The CoG Changes!

• Oblivious: Gathering two robots is not possible!

• No common direction or orientation: Gathering two robotsis not possible!

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Conclusion

Results on gathering

Gathering point robots

Scheduling Visibility Agreement Multiplicity Resultsrange in coordinate detection

Sync unlimited No No Solved [AOSY1999].ASync Any Yes No Solved [FPSW2001].ASync unlimited No Yes Not solvable for two

robots [P2007].ASync unlimited No Yes Solved for three and

four robots[CFPS2012].

Solved for more thanfour robots initially

(a)in bi-angularconfiguration.(b)not in anyregular n gon

ASync unlimited No No Not solvable[P2007,CRTU2015].

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Conclusion

Results on gathering

Gathering fat robots

Scheduling Agreement in Resultsco-ordinate

ASync No Solved for up tofour robots [CGP2009].

Async No Gathering any number oftransparent fat robots.

without collision [GM2010].Sync No Solved for any number

of robots [CDFHKKKKMHRSWWW2011].(randomized / considering.robots with identification

and communication power).Sync No Solved by simulation [BKF2012].

ASync Chirality Solved for anynumber of robots [AGM2012].

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Conclusion

Fault tolerant gathering of point robots

Earlier results

Scheduling Assumptions # of ResultsFaulty Robots

SSync Multiplicity 1 Solveddetection [AP2006].

ASync Strong Arbitrary SolvedMultiplicity for more thandetection 2 robots

and Chirality [BDT2012].

Recent extension

Scheduling Assumptions # of ResultsFaulty Robots

ASync One Arbitrary Solved foraxis any initial

agreement configurations [BGM2015].

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Conclusion

Fault tolerant gathering of point robots [BGM2015]

• A given robot configuration C on a 2-D plane.

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Outline of the algorithm

• Draw horizontal lines through each robots in C.

L1(C)L2(C)L3(C)L4(C)L5(C)

L6(C)

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Conclusion


Case-1: L1(C) contains one robot position.

L1(C)

L2(C)

L3(C)

L4(C)

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Conclusion



• Let ri be the robot position on L1(C).

L1(C)

L3(C)

L4(C)

L2(C)

ri

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Conclusion



• ri does not move.

L1(C)

L2(C)

L3(C)

L4(C)

ri

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Conclusion



• ri does not move.

• All other robots move towards ri along straight lines.

L1(C)

L2(C)

L3(C)

L4(C)

ri

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Case-1: Correctness.

• L1(C) always contains one robot position.

L1(C)

L2(C)

L3(C)

L4(C)

ri

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• L1(C) always contains one robot position.

• Wait free algorithm and hence can tolerate arbitrarynumber of faults.

L1(C)

L2(C)

L3(C)

L4(C)

ri

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Case-2: L1(C) contains more than one robot positions.

L1(C)

L2(C)

L3(C)

L4(C)

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Conclusion



• ri and rj be the two corner robot positions on L1(C).

L1(C)

L2(C)

L3(C)

L4(C)

ri rj

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Conclusion



• Draw the equilateral triangle 4riTrj .

ri rj

T

L1(C)

L2(C)

L3(C)

L4(C)

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Conclusion



• ri and rj move towards T .

ri rj

T

L1(C)

L2(C)

L3(C)

L4(C)

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• ri and rj move towards T .

• All other robots move towards the nearest robot among riand rj .

L1(C)

L2(C)

L3(C)

L4(C)

ri rj

T

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• ri and rj move towards T synchronously.

ri rj

T

L2(C′)

L3(C′)

L4(C′)

L5(C′)

L1(C′)

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• ri and rj move towards T asynchronously.

ri

rj

T

L3(C′)

L4(C′)

L5(C′)

L6(C′)

L2(C′)

L1(C′)

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• ri and rj move towards T asynchronously.

• Once ri reaches T , it becomes stationary.

ri

rj

T

L3(C′′)

L4(C′′)

L5(C′′)

L6(C′′)

L2(C′′)

L1(C′′)

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Future Scope

What happens if,

• the robots are not see through?

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Conclusion

Gathering under unequal visibility range [CGM2015]

Computational model

• Unequal Visibility Rangea, agreement on both axes.

aNo result reported till now considering unequal visibility range.

r1

r2

r4

r5

r3

• The robots can see finite unequal ranges (visibility ranges)around themselves.

• initially visibility is symmetric.

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Computational model

• Visibility graph is Strongly Edge Connected Graph (SECG)a

aA digraph G is SECG if (u, v) ∈ E → (v, u) ∈ E.

r1

r2

r4

r5

r3

• The robots are treated as the nodes of the graph.

• If two robots are mutually visible they are connected bytwo arcs.

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New result

The algorithm presented in ”P Flocchini, G. Prencipe, N.Santoro, and P. Widmayer. Gathering of asynchronous robotswith limited visibility. Theoretical Computer Science,337(1-3):147 - 168, 2005.” also works for limited nonuniformvisibility ranges with a slight modification.

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Features of the algorithm

• Let R = {r1 . . . , rn} be a finite set of robots.

• The robots do not know the total number of robots.

• Minimum visibility range ∆ > 0 known to all robots.

• initially G is SECG.

The algorithm assures

• G remains SECG throughout the algorithm.

• the robots gather in finite time.

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Move Down

r

r′

∆̄

Vr

V Cr

vr

• r sees robots only below on Vr(where Vr: vertical line throughr).

• r′ : is the robot nearest to r onVr.

• ∆̄ = Min(∆, Dist(r, r′))(whereMin(a, b): the minimumbetween a and b; Dist(a, b):thedistance between a and b.)

• Compute a point Tr on Vr suchthat Dist(r, Tr) = ∆̄

• r moves to Tr

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Move Right

r

r′

∆̄

Vr V ′r

V Cr

p

vr

• r sees robots only to its right

• r′ : the robot nearest to r andlies on the vertical line just nextto Vr.

• p : is the projection of r′ on Hr

(where Hr: horizontal line drawnthrough r).

• ∆̄ = Min(∆, Dist(r, p)).

• Compute a point Tr on Hr suchthat Dist(r, Tr) = ∆̄

• r moves to Tr

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Move Diagonal

rH

∆̄

r̄r′

V ′r

Vr

V Cr

β

β

B

C

A

vr

• When r sees robots both below on Vr andon its right

• r′ : the robot nearest to r on the verticalline just next to Vr

• B := Upper intersection point betweenV C(r) and Vr′(where V Cr: The circle centered at r)

• C := Lower intersection point betweenV C(r) and Vr′

• A := Point on Vr at distance vr below r

• 2β := Ar̂B

• If β < 60◦

• Rotate B around r such that β = 60◦

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Move Diagonal

rH

∆̄

r̄r′

V ′r

Vr

V Cr

β

β

B

C

A

vr

• Let B′ be the position of B afterrotation

• H :=The point on VB and onthe diagonal of the parallelogramwith sides rB and rA

• ∆̄ = Min(∆, Dist(r,H))

• Compute a point Tr along theray rH such thatDist(r, Tr) = ∆̄

• r moves to Tr

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Correctness of the algorithm

Lemma

GR remainsconnected duringdown movementsof any robotr ∈ R.

r

r′

∆̄

Vr

V Cr

vr

Lemma

Every internalchord of a trianglehas length less orequal to thelongest side of thetriangle.

Lemma

GR remainsconnected duringright movementsof any robotr ∈ R.

r

r′

∆̄

Vr V ′r

V Cr

p

vr

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rH

∆̄

r̄r′

V ′r

Vr

V Cr

β

β

B

C

A

vr

Lemma

GR remainsconnected duringdiagonalmovements of anyrobot r ∈ R.

• r′ and r̄ cannot move.

• As r movesdiagonallyDist(r, r′)and Dist(r, r̄)reduce.

• Mutual connectivity between r,r′ and r, r̄ remainsintact.

Theorem

The graph GR remains connected during the executionof the algorithm.

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Theorem

The robots in R will gather in finite timeVl VR

Lemma

Let VL be the left most vertical line,i.e., no robots lie to its left. Either (i)one of the robots on VL will leave, or(ii) all the robots on VL will begathered to the bottom-most roboton VL, in finite time.

Corollary

If there exists any robot at the right side of VL, then allrobots on VL will leave VL in finite time.

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Theorem

The robots in R will gather in finite time

Vl VR

Lemma

The robots in R will not cross VR(right most vertical line).

Lemma

Distance between VL and VR reducesby a finite amount in finite number ofmovements of the robots.

• Note: VL changes due to robots movement, but VR isfixed

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Theorem

The robots in R will gather in finite time

VlVR

Lemma

After a finite time there exists novertical line between VL and VR.

Lemma

All the robots in R will reach VR infinite time.

Lemma

If VL = VR, all the robots gather ondown most robot in finite time. 65 / 70


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Future Scope

• This algorithm is based on mutual visibility of the robotsand having agreement in the direction of X − Y .

• Is gathering possible if the robots are not mutually visible?

• Is gathering possible if the robots have agreement in oneaxis?

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Robots

KrishnenduMukopad-

hyaya

Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Gathering is still an open problem!

• In presence of obstacles

• Near gathering

• In graph(discrete plane)

• ...

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Robots

KrishnenduMukopad-

hyaya

Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Future Direction

• Obstructed Visibility [AFGSV2014, AFFSV2014].

• Unequal visibility range [CGM2015].

• Light model [DFPS2012, AFGSV2014]

• Flocking [CP2007]

• Designing Optimal/efficient Algorithms.

• ...

Motivation

What are the minimal requirements to solve a problem?

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Robots

KrishnenduMukopad-

hyaya

Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

Swarm robots moving in graph

Robots moving on graph → discrete movements

• Gathering in graph [ASN2014]: ring, tree, grid.

• Exploring a graph: tree [FIPS2008]

• ...

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Robots

KrishnenduMukopad-

hyaya

Introduction

Computationalmodel



Leader election

Circle formation

Gathering



Conclusion

References

• ”Distributed Computing by Oblivious Mobile Robots”,Flocchini, P. and Prencipe, G. and Santoro, N., Morgan &Claypool Publishers, Synthesis Lectures on DistributedComputing Theory, 2012.

• Thesis by Prencipe, G (2002),http://sbrinz.di.unipi.it/peppe/Articoli/TesiDottorato.pdf

Thank You

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Documents

Distributed Algorithms for Swarm Robotsigwoa/IGWOA_KM.pdf · Distributed Algorithms for Swarm Robots Krishnendu Mukopad-hyaya Introduction Computational model Examples of some problems