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Multi-Robot Task Scheduling with Ant Colony Optimization in Antarctic Environments

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Seokyoung Kim
Seokyoung Kim
Seokyoung Kim
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Heoncheol Lee
Heoncheol Lee
Heoncheol Lee
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This paper addresses the problem of multi-robot task scheduling in Antarctic environments. There are various algorithms for multi-robot task scheduling, but there is a risk in robot operation when applied in Antarctic environments. This paper proposes a practical multi-robot scheduling method using ant colony optimization in Antarctic environments. The proposed method was tested in both simulated and real Antarctic environments, and it was analyzed and compared with other existing algorithms. The improved performance of the proposed method was verified by finding more efficiently scheduled multiple paths with lower costs than the other algorithms.
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Sensors2023,23,751.https://doi.org/10.3390/s23020751www.mdpi.com/journal/sensors
Article
Multi‐RobotTaskSchedulingwithAntColonyOptimization
inAntarcticEnvironments
SeokyoungKimandHeoncheolLee*
DepartmentofITConvergenceEngineering,KumohNationalInstituteofTechnology,
Gumi‐si39177,RepublicofKorea
*Correspondence:hclee@kumoh.ac.kr;Tel.:+82‐54‐478‐7476
Abstract:Thispaperaddressestheproblemofmulti‐robottaskschedulinginAntarcticenviron‐
ments.Therearevariousalgorithmsformulti‐robottaskscheduling,butthereisariskinrobot
operationwhenappliedinAntarcticenvironments.Thispaperproposesapracticalmulti‐robot
schedulingmethodusingantcolonyoptimizationinAntarcticenvironments.Theproposedmethod
wastestedinbothsimulatedandrealAntarcticenvironments,anditwasanalyzedandcompared
withotherexistingalgorithms.Theimprovedperformanceoftheproposedmethodwasverifiedby
findingmoreefficientlyscheduledmultiplepathswithlowercoststhantheotheralgorithms.
Keywords:Antarcticenvironments;antcolonyoptimization;multi‐robottaskscheduling
1.Introduction
Duetothedevelopmentofrobottechnology,robotsareworkinginsteadofhumans
inmanyplaces.Robotshavetheadvantageofbeingabletoperformprecisetasksthat
humanscannotdo,andtasksthataredangerousforhumanstodo.Basedonthesead‐
vantages,robotsareusedinmanyplacessuchasinhomes,services,industry,medical
applications,andmilitaryapplications.Robotapplicationsusingasingleobject,suchasa
cleaningrobot,aguiderobot,andaprocessusingarobotarm,havebeencommercialized.
However,inthecaseofasinglerobot,thelimitationsareclear,suchaslowefficiencyor
unachievabletasks.Toovercomethis,multi‐robotsbegantobeintroduced,whichshowed
increasedworkefficiencyandmorediversemissionperformance.Inotherwords,using
multi‐robotsenablesefficientperformanceoftaskssuchasfastprocessingspeedandlarge
workload,butthisrequiresadvancedtechnology.Thisisbecauseasthenumberofrobots
increases,thecontrolstructureandcalculationtimeincreasedramatically.Forthisreason,
researchfieldsformulti‐robotssuchastaskallocation,coverage,andschedulinghave
beencreatedandarebeingstudiedsteadily.
Amongthem,schedulingisfortheefficientoperationoftherobotandaimstoreduce
thedrivingtimeoftherobot.Whentherearemultiplerobotsandmultipledestinations,
eachrobotisgivenanappropriatevisitordertominimizetherobot’straveldistanceand
reducedrivingtime.Thiscanbeseenasakindoftravelingsalesmanproblem(TSP)[1–
5].TheTSPistofindtheshortestpossibleroutefromagivensetofcities,visitingevery
cityexactlyonceandreturningtothestartingpoint.TheTSPisNP‐Hard,andtherearea
numberofalgorithmstosolvethisproblem.Representatively,therearebreadth‐first
search(BFS)anddepth‐firstsearch(DFS)[6].Thesearealgorithmsfortraversingor
searchingtreeorgraphdatastructures,whichguaranteestheminimumdistance,buthas
thedisadvantageofconsumingalotofresourceswhenthepathislongandnotguaran‐
teeingaproblem‐solvingtime.Nearestneighbor[7]isasimplealgorithmtorepeatvisit‐
ingthenearestcity.Thishastheadvantageofbeingeasytoimplementandfasttocalcu‐
late,butitdoesnotguaranteeminimumdistance.Thegeneticalgorithm(GA)[8–11],one
oftheheuristicalgorithms,guaranteesdistanceandtimeaccordingtothesettingofthe
Citation:Kim,S.;Lee,H.
Multi‐RobotTaskSchedulingwith
AntColonyOptimizationin
AntarcticEnvironments.
Sensors2023,23,751.
https://doi.org/10.3390/s23020751
AcademicEditor:AiguoSong
Received:5December2022
Revised:31December2022
Accepted:3January2023
Published:9January2023
Copyright:©2023bytheauthors.Li‐
censeeMDPI,Basel,Switzerland.
Thisarticleisanopenaccessarticle
distributedunderthetermsandcon‐
ditionsoftheCreativeCommonsAt‐
tribution(CCBY)license(https://cre‐
ativecommons.org/licenses/by/4.0/).
Sensors2023,23,7512of14
userparameter.Iftheuserparameterisproperlyset,thedistancecanbecalculatedwith
areasonablecalculationtime.Antcolonyoptimization(ACO)[12–20]isalsoaheuristic
algorithmthatwasconceivedinthewayantsreturnhomeinsearchoffood.Variousap‐
proacheshavebeentakentosolvetheTSPusingACO,whichalsomadeitpossibleto
obtaindistanceswithreasonablecomputationaltime.WhentheTSPisappliedtomulti‐robots,
itiscalledthemultipletravelingsalesmanproblem(MTSP)[21–28].TheMTSPisanoptimi‐
zationprobleminwhichmultiplesalesmenvisitalldestinationswithminimaldistance.Many
approacheshavebeentakentosolvetheMTSPbasedontheabovealgorithms.
However,theseTSPsolutionswillbecomemoredifficulttoimplementinextreme
environments.TheextremeenvironmentexaminedhereisAntarctica,aplacewithalarge
areaofabout14,000,000km2,verylowtemperatures,andvariousadverseconditionsin‐
cludingsnow,ice,andcrevasses.AntarcticaisthesouthernmostcontinentoftheEarth,
andisanattractiveunexploredregionwithenormousscientificvalue,fisheryresources,
andenergyresourcestocopewiththeEarth’sclimatechangeproblems.Todiscoverthis
valueofAntarctica,47countrieshavejoinedtheAntarcticTreatyandarefiercelycompet‐
ingforAntarcticresearch.ManyscientistsaretryingtostudyAntarctica,butinextreme
weather,crevassesmakeitdifficultforhumanstoexplorethepolarregions.Toovercome
this,scientistshavebeguntoresearchandintroduceunmannedautonomousdrivingro‐
bots.Theoperationofrobotsingeneralenvironmentssuchasroadsandindoorsisrela‐
tivelyfreefromtheaforementionedconstraints.Thegeneralenvironmentdoesnotmake
robotoperationdifficultbecausethefloorisrelativelyflatandnotslippery,andthereare
fewfatalobstaclessuchascrevasses.However,inAntarcticenvironments,slopingareas
suchashillsandmountains,slipperyfloorscausedbysnoworice,andcrevassesshould
beconsidered.TheColdRegionsResearchandEngineeringLab(CRREL)intheUnited
Stateshasdeveloped‘CoolRobot’[29]and‘Yeti’[30],autonomousvehiclesthatcanbe
operatedinpolarenvironments,andtheyareusedtocollectresearchdatainextremeareas
suchasAntarctica.However,theyshowdisadvantagesinenvironmentssuchassnowand
ice.Inaddition,theaveragespeedoftherobotisabout0.4to1.5m/s,whichisslow,andit
issomewhatdisadvantageousintimetoexplorethevastareaofAntarctica.Therefore,the
needforefficientexplorationworkusingmulti‐robotsratherthansinglerobotshas
emerged.Figure1describestheconceptofmulti‐robotscheduling.
Inthispaper,weproposeapracticalmulti‐robotschedulingmethodinAntarcticen‐
vironments.Thisallowsmulti‐robotstovisitallnodeswiththeshortestdistance.Thiscan
beseenasakindofMTSPproblem‐solving,butconsideringthespecificityofAntarctica,
theprocessofreturningtothestartingpointaftervisitingallnodeswasomitted.Inaddi‐
tion,stabledrivingcanberealizedbyavoidingsharpslopesbyreflectingAntarcticalti‐
tudeinformationinscheduling.Itcanalsoproducebetterresultswithreasonablecompu‐
tationaltime.Thecontributionsofthispaperareasfollows.
Tothebestofourknowledge,thisisthefirstapproachwhichsolvesthemulti‐robot
taskschedulingprobleminAntarcticenvironments.
Theperformanceofthemulti‐robottaskschedulingresultwastestedandevaluated
inbothsimulatedandrealAntarcticenvironments.
Thescheduledpathsbytheproposedmethodcanimprovetheefficiencyofoperating
multiplerobotsbyconsideringthecharacteristicsofrobotmovementinAntarctic
environments.
Theremainderofthispaperisorganizedasfollows.Section2describesconstraints
intheAntarcticenvironmentsandthenecessityforschedulingincludingconstraints,as
wellasthedefinitionofMTSPandproblemsofapplyingexistingalgorithmstoAntarctic
environments.Section3describesthecostfunctionandthestructureofACOusedfor
multi‐robotscheduling.Section4showstheexperimentalresultsandthecomparisonwith
othermethods.Finally,Section5istheconclusion.
Sensors2023,23,7513of14
Figure1.Theconceptofmulti‐robotschedulinginAntarcticenvironments.
2.ProblemDescription
Thispaperaddressestheproblemofmulti‐robotschedulinginAntarcticenviron‐
ments.First,wedefinethespecificityoftheAntarcticenvironmentanditsproblems.It
addressesproblemsthatcanbecausedbyextremelylowtemperatures,snowandiceen‐
vironments,andaltitudes.Apracticalschedulingmethodforovercomingtheseproblems
isdescribedlater.ThiscanbeseenasakindofMTSP,butconsideringthecharacteristics
ofAntarcticenvironments,itisassumedthatthemulti‐robotdoesnotreturntothestart‐
ingpointaftervisitingallnodes.
2.1.AntarcticEnvironments
AntarcticaisoneofthecoldestregionsonEarth,coveringanareaofabout14
14,000,000
km
2,ofwhich98%ismadeupofsnowandice.Inallregions,thetemperature
doesnotexceed0°C,andthelowesttemperatureis−89.2°C,whichisthecoldestarea.
Theseconditionsmakeitdifficulttooperatetherobot.Forexample,whenexposedtolow
temperatures,itcausesdamagetothebatteryandisbadforthechassisoftherobot.Ad‐
ditionally,thesnowandicefloorreducetherobot’sabilitytomove.Forstabledrivingin
snowandonicyterrain,itwillbenecessarytoavoidslopesinconsiderationofheight.
Thecrevasse,adeepcrackontheglacialsurface,isalsooneoftheobstaclesthatmustbe
avoided.Forstablerobotoperation,theseconstraintsshouldbeavoidedasmuchaspos‐
sible.Therefore,schedulinginAntarcticenvironmentsneedstoreflectelementsofthe
terrainaswellasdistanceinthecost.
2.2.DefinitionofMTSP
Inthispaper,thedefinitionofMTSPisasfollows.Themulti‐robotschedulingprob‐
lemisdefinedasvisitingagivenasetofnodes𝐂𝑐
,
𝑐
,𝑐
,… ,𝑐
,where 𝑛1, ⋯,𝑁
foreachrobot𝑟,withtheshortestdistance.Eachrobothasanumberofvisits𝐏
󰇝𝑝
,
𝑝
,𝑝
,… ,𝑝
󰇞,where 𝑟1, ⋯,𝑅.Itisdefinedasasingledepotifthereisonestarting
positionandamultipledepotiftherearemultiplestartingpositions.Inthispaper,asingle
depotisassumed.EachoftheRrobotslocatedinthesingledepotmustvisitoneormore
Sensors2023,23,7514of14
nodesandwillnotreturntothestartingposition.Eachrobothasatour𝐓,whichisde‐
scribedasfollows.𝐓󰇝𝑐󰇞,⋯, where 𝑐∈𝐂(1)
where𝑐isthenodevisitedbytherobot𝑟and𝑝isthetotalnumberofnodesthatthe
robot𝑟willvisit,calculatedasfollows.
𝑝
 𝑁(2)
Inthetour𝐓,whenthedistancebetweenthenodes𝑐and𝑐
is𝑑,thetotal
tourdistanceforthetour𝐓,isasfollows.
𝐷󰇛𝐓󰇜𝑑

 (3)
𝐓,,thetourwiththeminimumtotaltraveldistance,isdefinedasfollows.
𝐓, argmin
𝐓󰇛𝐷󰇛𝐓󰇜󰇜(4)
Then,thegoalistoobtainasetof𝐓,for𝑅robots.
𝐓 𝐓,,𝐓,,⋯,𝐓,(5)
Tominimizethedistance,itisrequiredtosetthenumberoftournodes𝑝foreach
robot𝑟andobtain𝐓throughanappropriatealgorithm.
2.3.TheProblemofApplyingtheExistingSchedulingAlgorithmtoAntarcticEnvironments
Variousalgorithmshavebeenstudiedtosolvethemulti‐robotschedulingproblem.
Amongthem,thenearestneighboralgorithmisasimplealgorithm,summarizedasfollows.
(1) Selectastartingpointforanycityandregisteritasavisitingnode.
(2) Movetotheunvisitednodewiththelowestcostandregisteritasthevisitednode.
(3) RepeatStep2ifthereisacitythatwasnotvisited.
Itissimpleandeffective.However,duetothegreedynatureoftheNNalgorithm,it
onlyseeksimmediatebenefits.Thus,itmissestheopportunitytomakelong‐termgains.
Thisleadstothecreationofabadpath.Whenscheduledbasedonthecostreflectingnot
onlythedistancebutalsothetopographicalelements,thesecharacteristicswillbere‐
vealedasdisadvantages.
ACOisalsooneofthealgorithmsforsolvingthemulti‐robotschedulingproblem.ACO
isanalgorithmthatsolvesproblemsbyexploringartificialants.Antshavearulethatthey
preferplaceswithalowcostandhighpheromones,whichissummarizedasfollows.
(1) Exploreants.
1.1 Anantselectsanodebytheprobability𝑝,whichisproportionaltotheamount
ofpheromonesandthecost.
(2) Whentheantsfinishtheirsearch,theyleavepheromonesinthepathoftheantthat
hasthelowestcost.
(3) Repeatasiteration.
(4) Afterthat,theantthatmovedtothelowestcostbecomesasolution.
InACO,pheromonesaswellascostareadditionallyconsidered.Moreover,proba‐
bilisticnodeexplorationallowsantstoexplorevariouspaths,whichgivesthemanoppor‐
tunitytochoosebetternodesinthelongterm.Thiseventuallymakesitpossibletofinda
betterpath.
ACOcaneasilycontrolthecostfunction,soitiseasytoevaluatefactorsotherthan
distance.Itisalsoimmediateandintuitivebecauseitreflectsthecosteachtimewhen
visitingthenodesonebyone.
Sensors2023,23,7515of14
3.ProposedMethod
3.1.Overview
Figure2isaflowchartoftheproposedalgorithm.ThisalgorithmisbasedonACO,
buttwomainfeaturesareadded.First,formulti‐robotscheduling,thenumberofnodes
eachrobotwillvisitisset.Thisisdeterminedbytheuserorautomaticallydividedbythe
numberofrobots.Then,pathsforeachrobot,anMTSPsolution,isgeneratedformulti‐
robotsusingACO.Afterpathsformulti‐robotsarecreatedusingtheproposedmethod,
eachrobotmovesaccordingtoitspath.Inthispaper,anewcostfunctionforACOispro‐
posedtoproperlyreflectthecharacteristicsofAntarcticenvironments.
Figure2.Theflowchartoftheproposedmethod.
3.2.CostFunction
Onlythedistancebetweennodeswasconsideredforcostfunctionusedintheexisting
ACO.ThisisdifficulttoreflectAntarcticenvironments.Theproposedcostfunctioncontains
elevationinformation.Intheexistingcostfunction,thedistancebetweennodes𝑝and𝑞is
calculatedastheEuclideandistance.However,thisisastraightdistance,whichbecomesin‐
accurateifaltitudeinformationisadded.Itisalsodifficulttomeasuretheexactdistance
Sensors2023,23,7516of14
betweennodes𝑝and𝑞includingaltitudeinformation.Thiswasovercomebyobtainingan
approximatedistancevaluebysamplingbetweennodes.Amethodofobtainingthedistance
betweennodes𝑝and𝑞includingthealtitudevalueisasfollows.
Thenode𝑝andthenode𝑞aresampled𝑘times,andthedistanceobtainedbydi‐
viding𝑑,by𝑘is𝑑,where𝑑,isthedistancebetweennodes𝑝and𝑞.Thealti‐
tudevalue𝐻,sampledbetweennode𝑝andnode𝑞isasfollows.
𝐻,󰇛ℎ,ℎ,ℎ,⋯ , ℎ,ℎ󰇜 (6)
whereℎistheheightofthe𝑘thsampledpoint.Thedifference𝑙ofthesampledheight
valueisasfollows.𝑙ℎℎ (7)
Theelevationdistance𝑑′,forreflectingthealtitudeinformationbetweennodesp
andqisdefinedasfollows.
𝑑′,𝑑𝑙

 (8)
ThevalueΘ,forreflectingthealtitudeinformationbetweennode𝑝andnode𝑞
isasfollows.
Θ,𝜃

 (9)
where𝜃istheanglebetweenthestraightlinebetweenℎandℎandthestraightline
paralleltothex‐axis.Finally,theproposedcostfunction𝑐,isasfollows.
𝑐,𝐴𝑑′,𝐵Θ, (10)
where𝐴and𝐵areweights,whichcanbearbitrarilydeterminedbytheuser.𝑑′,isthe
distanceofthecitytime,andΘ,isthealtitudevalueofthecitytime.Figure3showsthat
thecostfunctioniscalculatedbasedonthealtitudeinformationincludedattheedgebe‐
tweennodes,wherethecurveisheightinformationandthestraightredlineisastraight
lineconnectingpointssampledatregularintervals;thesumofthelengthsofthestraight
redlineis𝑑′,,andthesumoftheanglesisΘ,.
Figure3.Exampleofcalculatingtheproposedcostfunctionbetweennodes𝑝and𝑞.
Sensors2023,23,7517of14
3.3.AntColonyOptimization
TherearevarioustypesofACOs;amongthem,AntColonySystem(ACS)[13]was
used.ACSisanimprovedalgorithmbyaddingseveralprocessestotheexistingAntSys‐
tem(AS)[14–16].Basedonthis,multi‐robotschedulingusingcostfunctionadaptedto
Antarcticenvironmentswasimplemented.
First,itistherandom‐proportionalrulethatantkvisitsfromnode𝑝tonode𝑞.
𝜔
󰇱𝜏𝜂
∑𝜏𝜂
∉,𝑖𝑓 𝑔∉𝑉
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(11)
where𝜏isthepheromonesand𝜂istheimportanceoftheedge,whichistheinverseof
thecost.ThecostisthevalueobtainedusingEquation(10).𝑉isthesetofnodesvisited
byant𝑘.𝛼isaparameterthatdeterminestheimportanceofpheromones,and𝛽isa
parameterthatdeterminestheimportanceofedgecost.Here,thestatetransitionruleof
ACSisappliedasfollows.
𝑠𝑎𝑟𝑔𝑚𝑎𝑥∉󰇝𝜏𝜂󰇞,𝑖𝑓 𝑧𝑧
𝑆, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(12)
where𝑆isarandomvariabledeterminedbyEquation(11).Itisaruleaboutantschoosing
apheromone‐richpath.Iftherandomnumber𝑧 󰇛0𝑧1󰇜islessthan𝑧,antschoose
thepathwherethepheromonelevelishigh,andifnot,itfollowstherandom‐proportional
ruleoftheAS.Thispreventsfallingintothelocaloptimalsolution.Random‐proportional
rulesandthestatetransitionrulesareapplied,andlocalparentupdateisperformedac‐
cordingtoEquation(13)whenevervisitinganode.
𝜏 󰇛1𝜑󰇜𝜏𝜑𝜏 (13)
where0𝜑1isapheromonedecayparameter.𝜏isainitialvalueofpheromone;it
isusually𝜏1/𝑛𝑐.𝑐isthenumberofcities,and𝑐isthecostcalculatedbythe
nearestneighbor.Thisallowsallantstobeaffectedbypheromonesinrealtimeandavoid
localoptimums.
Whenallantsgenerateatour,globalpheromoneupdateisperformedthroughEqua‐
tions(14)and(15).𝜏 󰇛1𝜌󰇜𝜏 Δ𝜏
 (14)
Δ𝜏
 1/𝑐,𝑖𝑓 𝑏𝑒𝑠𝑡 𝑎𝑛𝑡 𝑡𝑟𝑎𝑣𝑒𝑙𝑠 𝑜𝑛 𝑛𝑜𝑑𝑒 𝑝,𝑞
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (15)
where𝑐isthebestant’stour.Thisincreasestheprobabilityofexploringabetterpath
inthenextiterationbyaccumulatingpheromonesalongthetourofthebestsolution.
Theproposedmethodappropriatelydividedthenumberofnodessothatmulti‐ro‐
botscanperformTSP.Althoughtheusermaydeterminethenumberofnodestobevisited
bytherobot,itisbasicallyimplementedbydividingthenumberofnodesbythenumber
ofrobots.Algorithm1isapseudo‐codefortheproposedmethod.

Sensors2023,23,7518of14
Algorithm1Multi‐robotschedulingalgorithminAntarcticEnvironments
1:Initializethemultirobot’stour
T
2:Setnumberofnodesthattherobotwillvisit
N
anditerationI
3:fori←Ndo
4:for
j
← Ido:
5:foreachantdo
6:Buildasolutionaccordingtothenumberofnodesi
7:Updatelocalpheromone
8:endfor
9:Updateglobalpheromone
10:endfor
11:Appendbestant’stourto
T
12:endfor
13:returnMultirobot’stourT
4.Results
4.1.ResultsinSimulationEnvironments
SimulationswereperformedtocomparetheproposedmethodwithNNandGA.
Theyshowaspecificperformancedifferencebycomparingtheelevationdistance.The
elevationdistanceusesthevalueaccordingtoEquation(8)accordingtothex,y,andz
coordinatesofthenodeandtheedge.ThesimulationswereperformedinPython3.9.7
andtheresultswerevisualizedusingmatplotlib.Theenvironmentwithinthesimulation
isavirtual3Dspace,whichis1000×1000×500pixels.Inordertorealizetheheightinthe
virtualspace,virtualhillsAandBaccordingtothenormaldistributionweregenerated
usingtheprobabilitydensityfunction.ThenormaldistributionfunctionvalueofhillAis
asfollows:σ75,μ20.Theheightis400.Thenormaldistributionfunctionvalueof
hillBisasfollows:σ100,μ15.Theheightis200.Thelocationofthenodeswas
randomlyset,andsimulationswereperformedon20,30,and40nodes.Figure4showsavir‐
tualspace,where(a)isthespaceviewedfromtheside,and(b)isthespaceviewedvertically.
(a)(b)
Figure4.(a)Thesideviewofthesimulationenvironment.(b)Theverticalviewofthesimulation
environment.AandBarethehillsofthesimulationenvironment.
Sensors2023,23,7519of14
Fortheelevationdistancecomparisonwiththeproposedmethod,thenearestneigh‐
boralgorithmandthegeneticalgorithmwereperformed.Thecostfunctionsofthepro‐
posedmethod,NN,andGAweredefinedaccordingtoEquation(10),andtheparameter
valueswereasfollows: 𝐴10, 𝐵15.TheparametervaluesofGAwereasfollows:
𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒0.05, 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛50, 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛300,selectionoperatorwastour‐
nament,crossoveroperatorwastwo‐pointcrossoverandelitismwasapplied.andelitism
wasapplied.TheparametervaluesofACOintheproposedmethodwereasfollows:
𝑎𝑛𝑡𝑠40, 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛20, 𝛼2, 𝛽𝑏,𝜑0.1, 𝜌0.05, 𝑧
0.5,𝑎𝑛𝑡𝑠isthenumberof
ants,𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛isthenumberofiterations.Figures5–7aretheresultsofNN,GA,andthe
proposedmethodinthesimulationenvironment,andTable1isacomparisontableofthe
elevationdistance.Figure8isacomparisonchartoftheelevationdistance.

(a)(b)(c)
Figure5.Simulationresultsofthenearestneighborinvirtualspacefor:(a)20nodes,(b)30nodes,
(c)40nodes.Thereddotisthestartingpointandthebluedotsarethenodestovisit.Eachcolorline
isthepathofeachrobot.

(a)(b)(c)
Figure6.Simulationresultsofthegeneticalgorithminvirtualspacefor:(a)20nodes,(b)30nodes,
(c)40nodes.
Sensors2023,23,75110of14

(a)(b)(c)
Figure7.Simulationresultsoftheproposedmethodinvirtualspacefor:(a)20nodes,(b)30nodes,
(c)40nodes
Figure8.Theelevationdistancecomparisoninsimulationenvironments.
Table1.Theelevationdistancecomparisonresultsofthesimulation.
PartNode20Node30Node40
NearestNeighbor5476.986255.677943.39
GeneticAlgorithm5729.226348.658335.68
ProposedMethod5584.365784.936484.76
Regardingtheresultsinsimulationenvironments,asshowninTable1,NNandthe
proposedmethodshowedsimilarresultsfor20nodes.However,asthenumberofnodes
increased,theproposedmethodshowedashorterelevationdistance.GAshowedlow
performanceinalloftheresults.NNisshorterincomputationaltime,butrealtimedoes
notneedtobeguaranteed,sotheproposedmethodisreasonablebygeneratingshorter
andmorestablepathswithlessthan10s.
4.2.ResultsinRealAntarcticEnvironments
SimulationsinAntarcticenvironmentswereperformedtocomparetheproposed
methodwithNNandGA.Asdescribedabove,specificperformancedifferencesarepre‐
sentedbycomparingtheelevationdistance.
Inthesimulation,theAntarcticenvironmentwaslocatedat74°37.4’S,164°13.7’E,
andnodeswererandomlysetnearby.Thelatitudeandlongitudevaluesofarbitrarynodes
Sensors2023,23,75111of14
wereextractedfromGoogleEarth.Thedistancebetweennodeswasobtainedusingthe
HaversineFormula.Thealtitudeinformationobtainedthealtitudevaluesofnodesand
edgesusingtheGoogleMapsAPI.Thealtitudevaluesfortheedgesweresampled500
timesatthesameinterval.Forperformancecomparisonwiththeproposedmethod,the
nearestneighboralgorithmandthegeneticalgorithmwereperformed.Thecostfunctions
oftheproposedmethod,NN,andGAweredefinedaccordingtoEquation(10),andthe
parametervalueswereasfollows: 𝐴3, 𝐵2.TheparametervaluesofGAwereasfol‐
lows:𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑡𝑜𝑢𝑟𝑛𝑎𝑚𝑒𝑛𝑡,𝐶𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟𝑡𝑤𝑜𝑝𝑜𝑖𝑛𝑡 𝑐𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟,𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
0.05, 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛50, 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛300,andelitismisapplied.Theparametervaluesof
ACOintheproposedmethodwereasfollows:𝑎𝑛𝑡𝑠40, 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛20, 𝛼2, 𝛽
𝑏,𝜑0.1, 𝜌0.05, 𝑧
0.5,𝑎𝑛𝑡𝑠isthenumberofants,𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛isthenumberofiter‐
ations.Figures9–11arethesimulationresultsofNN,GA,andtheproposedmethodin
theAntarcticenvironments,respectively,andTable2isacomparisontableoftheeleva‐
tiondistance.Figure12isacomparisonchartoftheelevationdistance.
RegardingtheresultsinAntarcticenvironments.AsshowninTable2,forallcases,
theACOgeneratedashorterpath,especiallywhenthenumberofnodeswasmorethan
20,showingbetterperformance.AlthoughNNhadashortercomputationaltime,thepro‐
posedmethodwaslessthan5s,whichcanbeconsideredreasonableifitgeneratesshorter
andmorestablepaths.
(a)(b)(c)
Figure9.SimulationresultsofthenearestneighborinAntarcticenvironmentsfor:(a)10nodes,(b)
20nodes,(c)30nodes.Thereddotisthestartingpointandthebluedotsarethenodestovisit.Each
colorlineisthepathofeachrobot.
(a)(b)(c)
Figure10.SimulationresultsofthegeneticalgorithminAntarcticenvironmentsfor:(a)10nodes,
(b)20nodes,(c)30nodes.
Sensors2023,23,75112of14
(a)(b)(c)
Figure11.SimulationresultsoftheproposedmethodinAntarcticenvironmentsfor:(a)10nodes,
(b)20nodes,(c)30nodes
Table2.TheelevationdistancecomparisonresultsofrealAntarcticenvironments.
PartNode10Node20Node30
NearestNeighbor82.01km99.89km145.67km
GeneticAlgorithm79.41km93.93km141.36km
ProposedMethod78.17km82.95km122.78km
Figure12.TheelevationdistancecomparisoninAntarcticenvironments.
5.Conclusions
Thispaperaddressestheproblemofpracticalmulti‐robottaskschedulinginAntarc‐
ticenvironments.WeanalyzedthedifficultiesofrobotoperationinAntarcticaandpresent
asolution.Forstablerobotoperation,ACOwithanovelcostfunctionincludingaltitude
informationisproposed.Theproposedmethodcreatesapaththatavoidssteepslopes,
enablingstablerobotoperationinAntarcticenvironmentsconsistingofsnowandice.Fur‐
thermore,thecomparisonoftheresultswiththenearestneighboralgorithmshowsthat
theproposedmethodgeneratesshorterpaths,enablingefficientscheduling.However,as
mentionedearlier,thereareanumberofconstraintsintheAntarcticenvironment,such
asaltitude,snow,ice,crevasses,windspeed,andlimitedcommunications.Inthispaper,
onlyafewconstraintsareconsidered.Variousfactorsmustbeconsideredformoreeffi‐
cientrobotoperation.Inthefuture,schedulingwillbecarriedoutconsideringvarious
constraintswhileincludingaltitudeinformation.Stableandefficientrobotoperationwill
bepossiblebyfurtherreflectingthefactorsofAntarcticenvironments.
60.00
80.00
100.00
120.00
140.00
160.00
Node10 Node20 Node30
ElevationDistance
NearestNeighbor GeneticAlgorithm Proposedmethod
Sensors2023,23,75113of14
AuthorContributions:Conceptualization,H.L.;Methodology,S.K.;Writing—originaldraft,S.K.;
Writing—review&editing,H.L.Allauthorshavereadandagreedtothepublishedversionofthe
manuscript.
Funding:Thisworkwassupportedinpartbytheprojecttitled‘ResearchonCo‐OperativeMobile
RobotSystemTechnologyforPolarRegionDevelopmentandExploration’,fundedbytheKorean
MinistryofTrade,Industry,andEnergy(1525011633);inpartbytheGovernment‐wideR&DFund
forInfectionsDiseaseResearch(GFID),fundedbytheMinistryoftheInteriorandSafety,Republic
ofKorea(grantnumber:20014854);andinpartbytheNationalResearchFoundationofKorea(NRF)
grantfundedbytheKoreagovernment(MSIT)(no.2021R1F1A1064358).
InstitutionalReviewBoardStatement:Notapplicable.
InformedConsentStatement:Notapplicable.
DataAvailabilityStatement:Notapplicable.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
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