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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
General characteristics of swarm robots
• Point/Unit Disc
• Autonomous
• Identical
• No message passing
• Sense surroundings
• Move on the 2D plane
• Limited computational power
4 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Execute wait-look-compute-move cycle.
In wait state robots do nothing.
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Execute wait-look-compute-move cycle.
Look
Rv
r
• Rv (visibility range) can be limited or unlimited
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Execute wait-look-compute-move cycle.
t
Computer
• r computes its destination t
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Execute wait-look-compute-move cycle.
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Execute wait-look-compute-move cyclesemi-synchronously.
• An arbitrary set of robots look at the same time.
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Agreement on co-ordinate system.
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Agreement on co-ordinate system.
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Computational model
• Agreement on co-ordinate system.
XY
r2
XY
r4X
Y
r1
Y
r5
X
X
Y
r3
• Robots having different SoD and different chirality
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Examples of some problems
• Arbitrary pattern formation [FPS2008]
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Geometric characterization for pattern formation
Symmetric PatternsLine of Symmetry
Line of Symmetry
(a) (b) (c)
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Arbitrary pattern formation
Asymmetric Patterns
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Arbitrary pattern formation
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Examples of some problems
• Leader Election [DPV2010]
r1
r2
r3 r4
r5
L
• The robots elect r1 as their leader.
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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.
28 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Examples of some problems
Gathering (point robots)
• Gathering [P2007] or Convergence [CP2006]
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Examples of some problems
Gathering (fat robots)
• Gathering Fat Robots [CGP2009].
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Fault tolerant gathering of point robots [BGM2015]
• A given robot configuration C on a 2-D plane.
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: L1(C) contains one robot position.
L1(C)
L2(C)
L3(C)
L4(C)
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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: L1(C) contains one robot position.
• Let ri be the robot position on L1(C).
L1(C)
L3(C)
L4(C)
L2(C)
ri
39 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: L1(C) contains one robot position.
• ri does not move.
L1(C)
L2(C)
L3(C)
L4(C)
ri
40 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: L1(C) contains one robot position.
• ri does not move.
• All other robots move towards ri along straight lines.
L1(C)
L2(C)
L3(C)
L4(C)
ri
41 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: Correctness.
• L1(C) always contains one robot position.
L1(C)
L2(C)
L3(C)
L4(C)
ri
42 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-1: Correctness.
• 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
43 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: L1(C) contains more than one robot positions.
L1(C)
L2(C)
L3(C)
L4(C)
44 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: L1(C) contains more than one robot positions.
• ri and rj be the two corner robot positions on L1(C).
L1(C)
L2(C)
L3(C)
L4(C)
ri rj
45 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: L1(C) contains more than one robot positions.
• Draw the equilateral triangle 4riTrj .
ri rj
T
L1(C)
L2(C)
L3(C)
L4(C)
46 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: L1(C) contains more than one robot positions.
• ri and rj move towards T .
ri rj
T
L1(C)
L2(C)
L3(C)
L4(C)
47 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: L1(C) contains more than one robot positions.
• 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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: Correctness.
• ri and rj move towards T synchronously.
ri rj
T
L2(C′)
L3(C′)
L4(C′)
L5(C′)
L1(C′)
49 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: Correctness.
• ri and rj move towards T asynchronously.
ri
rj
T
L3(C′)
L4(C′)
L5(C′)
L6(C′)
L2(C′)
L1(C′)
50 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
Case-2: Correctness.
• 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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Future Scope
What happens if,
• the robots are not see through?
52 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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.
53 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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.
56 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
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
57 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
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
58 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
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◦
59 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Outline of the algorithm
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
60 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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
61 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Correctness of the algorithm
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.
62 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Correctness of the algorithm
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.
63 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Correctness of the algorithm
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
64 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Correctness of the algorithm
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
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
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?
66 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Gathering is still an open problem!
• In presence of obstacles
• Near gathering
• In graph(discrete plane)
• ...
67 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
Conclusion
Swarm robots moving in graph
Robots moving on graph → discrete movements
• Gathering in graph [ASN2014]: ring, tree, grid.
• Exploring a graph: tree [FIPS2008]
• ...
69 / 70
DistributedAlgorithmsfor Swarm
Robots
KrishnenduMukopad-
hyaya
Introduction
Computationalmodel
Examples ofsomeproblems
Arbitrarypatternformation
Leader election
Circle formation
Gathering
Fault tolerantgathering ofpoint robots
Gatheringunder unequalvisibility range
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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