Robotic Adversarial Coverage of Known Environments

39 Pages·2015·0.63 MB·English

TheInternationalJournalofRobotics
Research
Robotic Adversarial Coverage of Known
0(0):1–39
©TheAuthor(s)2015
Environments
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RoiYehoshua∗,NoaAgmonandGalA.Kaminka
ComputerScienceDepartment,BarIlanUniversity,Israel
Abstract
Coverage is a fundamental problem in robotics, where one or more robots are required to visit each point in a target
areaatleastonce.Mostpreviousworkconcentratedonfindingacoveragepaththatminimizesthecoveragetime.Inthis
paperweconsideranewandmoregeneralversionoftheproblem:adversarialcoverage.Here,therobotoperatesinan
environmentthatcontainsthreatsthatmightstoptherobot.Theobjectiveistocoverthetargetareaasquicklyaspossible,
whileminimizingtheprobabilitythattherobotwillbestoppedbeforecompletingthecoverage.Thisversionoftheproblem
hasmanyreal-worldapplications,fromperformingcoveragemissionsinhazardousfieldssuchasnuclearpowerplants,to
surveillanceofenemyforcesinthebattlefieldandfielddemining.Inthispaperwediscusstheofflineversionofadversarial
coverage,inwhichthemapofthreatsisgiventotherobotinadvance.First,weformallydefinetheadversarialcoverage
problem,andpresentdifferentoptimizationcriteriausedforevaluationofcoveragealgorithmsinadversarialenvironments.
WeshowthatfindinganoptimalsolutiontotheadversarialcoverageproblemisNP-Hard.Wethereforesuggesttwoheuristic
algorithms:STAC,aspanning-treebasedcoveragealgorithm,andGAC,whichfollowsagreedyapproach.Weestablish
theoreticalboundsonthetotalriskinvolvedinthecoveragepathscreatedbythesealgorithmsandontheirlengths.Lastly,
wecomparetheeffectivenessofthesetwoalgorithmsinvarioustypesofenvironmentsandsettings.
1. Introduction
Areacoverageisanimportanttaskformobilerobots,withmanyreal-worldapplicationsinvariousdomains,fromautomatic
floorcleaning(ColegraveandBranch,1994)andcoatinginsupermarkets(Endresetal.,1998)andtrainstations(Yaguchi,
1996),tohumanitarianmissionssuchassearchandrescueandfielddemining(NicoudandHabib,1995).Inthese,arobotis
givenaboundedwork-area,possiblycontainingobstacles,andisrequiredtovisiteverypartofitasefficientlyaspossible.
All previous studies of the coverage problem dealt with non-adversarial settings, where nothing in the environment is
hinderingtherobot’stask.However,inmanyoccasions,robotsandautonomousagentsneedtoperformcoveragemissions
inhazardousenvironments,suchasoperationsinnuclearpowerplants,planetaryexploration,demining,andsurveillance
ofenemyforcesinthebattlefield.
∗Correspondingauthor;e-mail:[email protected]
2 TheInternationalJournalofRoboticsResearch0(0)
Hence,ourworkaddressestheproblemofadversarialcoverage,thetaskofvisitingeverypointinagiventerrainwithout
beingdetectedordamagedbyanadversary.Eachpointinthearea,whichisnotoccupiedbyanobstacle,isassociatedwith
aprobabilityoftherobotbeingstoppedatthatpoint.Theobjectiveoftherobotistocovertheentiretargetarea(including
thethreatpoints)asquicklyaspossiblewhilemaximizingitsownsafety.Thisisapropergeneralizationofnon-adversarial
coverage,whichisthesameasadversarialcoverageinenvironmentswithnoadversarialpresence,orwhentheobjective
takesintoaccountonlythecoveragetimeandignorestheriskfactor.
Inthispaperwediscusstheofflineversionofthisproblem,inwhichthemapofthreatsisgiveninadvance,therefore
thecoveragepathoftherobotcanbedeterminedpriortoitsmovement.Weformallydefinetheofflineadversarialcoverage
problem and show that it is NP-complete. We then propose two algorithms for solving it heuristically in polynomial
time.TheSTAC(Spanning-TreeAdversarialCoverage)algorithmsplitsthetargetareaintoconnectedareasofsafeand
dangerous cells, and then it covers the safe areas before moving to the dangerous ones. The GAC (Greedy Adversarial
Coverage)algorithmfollowsagreedyapproach,whichleadstherobotfromitscurrentlocationtothenearestsafestlocation
whichhasnotbeencoveredyet,andshowempirically,insimulations,thatwhileSTACtendstoachievehigherexpected
coverage,GACproducescoveragepathswithloweraccumulatedrisk.Weprovideoptimalityboundsonbothalgorithms,
andshowthatSTACgeneratescoveragepathswhicharenearly-optimalwithrespecttotheexpectedcoveragepercentage,
and are also close to optimal with respect to the probability of completing the coverage in environments that consist of
contiguousareasofthreats.
Thepaperisorganizedasfollows.Section2presentsbackgroundandrelatedwork.Section3definestheadversarial
coverageproblemanddiscussesitscomplexity.Sections4and5presenttheSTACandGACalgorithms,respectively,and
analyzethemtheoretically.Section6presentsexperimentalresultsevaluatingthesealgorithms.Section7concludeswith
asummaryandbriefoverviewoffuturework.Forconcisenessandbetterreadability,someoftheproofsaredeferredto
appendices.
2. BackgroundandRelatedWork
The problem of single and multi-robot coverage has been extensively discussed in the literature and many approaches
to coverage path planning have been developed. Galceran and Carreras (Galceran and Carreras, 2013) provide a recent
exhaustivesurveyofcoveragepathplanning(CPP)methods,whichareclassifiedaccordingtovariouscriteria.
First, most coverage algorithms decompose the target space in subregions (called cells) to achieve coverage, thus
coverage algorithms can be classified according to the type of decomposition used. The survey (Galceran and Carreras,
2013)distinguishesbetweenapproximatecellulardecomposition,wherethefreespaceisapproximatelycoveredbyagrid
ofequally-shapedcells,andexactdecomposition,wherethefreespaceisdecomposedtoasetofregions,whoseunionfills
theentireareaexactly.Weuseapproximatedecompositioninthispaper.
Independently,coveragealgorithmscanbeclassifiedaseitherofflineoronline.Offlinealgorithmsrelyonlyonstationary
information, and the map of the environment is assumed to be known in advance. Conversely, online algorithms do not
assume full prior knowledge of the environment to be covered and utilize real-time sensor measurements to sweep the
targetspace.Inthispaperwefocusonofflinecoverage.
Thecoverageproblemisanalogoustothetravelingsalesmanproblem,whichisNP-completeevenonsimplegraphs
suchasgridgraphs(Papadimitriou,1977).However,itispossibletofindsolutionstothecoverageproblemthatareclose
tooptimalinpolynomialorevenlineartimethroughheuristicsandabstractions(e.g.,(Arkinetal.,2000),(Gabrielyand
Rimon,2003),(Grignietal.,1995),(Xuetal.,2011)).
Our own approach builds on earlier work in coverage; specifically, on the Spanning Tree Coverage (STC) family of
algorithms,firstintroducedbyGabrielyandRimon(GabrielyandRimon,2001)forsinglerobots,andextendedtomultiple
robotsbyHazonandKaminka(HazonandKaminka,2008).Thekeyideainthisfamilyofalgorithmsistoapproximatea
Yehoshua,AgmonandKaminka 3
two-dimensionalworkareausingagrid,suchthataHamiltoniancycleisguaranteedtoexistthroughthegrid,whichcan
befoundbygeneratingaspanningtreeinthegridgraph.TheoriginalSTCalgorithmgeneratesanoptimalcoveragepath
oftheenvironment,whichvisitseverycellinthegridexactlyonce.However,ittreatedcellswhicharepartiallyoccupied
byobstaclesascompletelyblocked.Inafollowuppaper(GabrielyandRimon,2003),GabrielyandRimonhaverefined
this algorithm to the Spiral Spanning Tree Coverage (Spiral-STC), which partially removes this limitation and provides
close-to-optimalcoveragepathsinauniformgridbasedterrain.
Theideawasbroadenedforamulti-robotsystembyHazonandKaminkainthefamilyofMultiRobotSpanningTree
Coverage (MSTC) algorithms. Their solution, along with decreasing the total coverage time, achieved robustness in the
sensethataslongasonerobotworksproperly,thecoverageoftheterrainisguaranteed.Theyhavealsoshownthatinmulti
robotteamsredundancymightbenecessaryformoreefficiency.Wealsoacceptredundantcoverage,butforsafety.
Elmaliach et al. (Elmaliach et al., 2009) extended this representation such that a non-uniform cost is attached to a
movementbetweentwoadjacentcellsinthegrid,andprovidedanalgorithmthatfindsthecyclicpathcoveringtheterrain
with minimum total cost. Conversely, in our work there is a uniform cost attached to each movement in the grid, and
additionallythereisaprobabilityofsurvivabilityassociatedwitheachcellinthegrid.Webelievethatourmethodscan
alsobeeasilyextendedtothecaseofnon-uniformmovementcosts.
Zelinskyetal.(Zelinskyetal.,1993)presentedthewavefrontalgorithm,whichisanothergrid-basedcoveragemethod.
Themethodrequiresastartcellandagoalcell.Adistancetransformthatpropagatesawavefrontfromgoaltostartisused
toassignaspecificnumbertoeachgridelement.Oncethedistancetransformiscalculated,acoveragepathcanbefoundby
startingonthestartcellandselectingtheneighboringcellwiththehighestlabelthatisunvisited.Thealgorithmissimple
andeasytoimplement.However,itdoesnotprovideanyguaranteesonthecoveragetime.Italsorequiresadefinitionofa
goalcell,thatisnotneededbythepreviouslymentionedcoveragemethods.
Papersintheroboticliteraturethattakeintoaccountthepresenceofanadversaryinclude(Bortoff,2000),(Likhachev
and Stentz, 2007), (Zabarankin et al., 2002). These present algorithms and methods for risk avoidance. These works
examine the path planning problem of a single robot, in order to bypass and avoid the adversary’s threats. In the patrol
problem(Elmaliachetal.,2009),amulti-robotteamneedstopatrolaroundaclosedareawiththeexistenceofanadversary
attemptingtopenetrateintothearea.Thepatrolproblemresemblesthecoverageprobleminthesensethatbothrequirethe
robotorgroupofrobotstovisitallpointsinthegiventerrain.However,whilecoverageseekstominimizethenumberof
visitstoeachpoint(ideally,visitingitonlyonce),patrollingoftenseekstomaximizeit(whilestillvisitingallpoints).
The adversarial coverage problem discussed in this paper is unique in the sense that the adversarial nature of the
environment is expressed in the possibility of physically harming the robot (rather than reducing the utility of a robotic
team,asin(HazonandKaminka,2008)).Ithasanintrinsiccomplexitythatisnottypicaltotheotherproblemsmentioned,
since it presents a delicate tradeoff between minimizing the accumulated risk and minimizing the total coverage time.
Tryingtominimizetheriskinvolvedinthecoveragepathmeansmakingsomeredundantsteps,whichinturncanmake
thecoveragepathlonger,andthusincreasetheriskinvolved,aswellasincreasethecoveragetime.
OtheroptimizationproblemsrelatedtoadversarialcoverageincludetheCanadianTravellerProblem(CTP)(Papadim-
itriou and Yannakakis, 1989), in which the objective is to find the expected shortest path between two nodes in a
partially-observable graph, where some edges may be non-traversable. In contrast, here the graph is fully-observable
andtheagentmustvisiteverynodeinthegraph(someofthemmaystoptherobot).
Theofflineadversarialcoverageproblemwasformallydefinedinourrecentstudy(Yehoshuaetal.,2013).Therewe
proposedasimplisticheuristicalgorithmthatgeneratesacoveragepathwhichtriestominimizeacostcomposedofboth
thesurvivabilityoftherobotandthecoveragepathlength.However,theheuristicalgorithmworkedonlyforobstacle-free
areas,andwithoutanyguarantees,oranalysisoftheproblemcomplexity.
In a follow-up paper (Yehoshua et al., 2014) we have addressed a more specific version of the adversarial coverage
problem,namely,findingthesafestcoveragepath.Therewesuggestedtwoheuristicalgorithms:STAC,aspanning-tree
4 TheInternationalJournalofRoboticsResearch0(0)
based coverage algorithm, and GSAC, which follows a greedy approach. We have shown that these algorithms produce
close to optimalsolutions in polynomial time. However,the heuristic algorithms couldhandle only one levelof threats,
i.e.,theenvironmentcouldcontaineithersafeordangerousareas,inadditiontoobstacles.
Inthispaper,wemakethefollowingcontributionscomparedtopreviousworks:
1.Weextendtheheuristicalgorithmstohandlemultiplelevelsofthreats.Weanalyzethetotalriskandthecoveragetime
ofthepathsgeneratedbythealgorithmsasafunctionofthenumberofthreatlevels.
2.Weaddressthegeneraladversarialcoverageproblemoffindingacoveragepaththattakesintoaccountboththerisk
andthecoveragetime.Weanalyzeitscomplexity,andshowhowtheheuristicalgorithmscanbemodifiedtohandle
thegeneralcase.
3.Weevaluatethemodifiedalgorithmsonanewvarietyofenvironmenttypesandsettings.
3. TheAdversarialCoverageProblemFormulation
Theadversarialcoverageproblemcanbedefinedasfollows.WearegivenamapofatargetareaT,whichcontainsobstacles
andalsothreats,whichmaystoptherobot.WeassumethatT canbedecomposedintoaregularsquaregridwithncells.
LetusdenotethisgridbyG.Gcontainstwotypesofcells:freecellsandcellsthatareoccupiedbyobstacles.Someof
thefreecellscontainthreats.Incontrasttoobstacleswhichtherobotcannotgothrough,threatlocationsareplacesthat
therobotmustvisit,butmightbestoppedat.Eachfreecelliisassociatedwithathreatprobabilityp ,whichmeasuresthe
i
likelihoodthatathreatinthatcellwillstoptherobot.Wedefinesafecellsascellswherethethreatprobabilityisp = 0,
i
whiledangerouscellsarethosewherep >0.Therobotcanmovecontinuously,inthefourbasicdirections(North,South,
i
East,West),andcanlocateitselfwithinthework-areatowithinaspecificcell.
Figure1showsanexampleworldmapofsize20×20.Obstaclesarerepresentedbyblackcells,safecellsarecolored
white,anddangerouscellsarerepresentedby5differentshadesofthesamecolor(appearsaspurpleinthepaper’sonline
versionandgrayinitsprintedversion).Darkershadesrepresenthighervaluesofp (moredangerousareas).
i
Fig.1. Anexamplemapoftheworld.Darkershadesrepresentmoredangerousareas.
Thesurvivabilityoftherobotcanbemeasuredintwodifferentways,whichimplytwodifferentobjectivefunctions:
1.Thetotalaccumulatedriskalongthecoveragepath(i.e.,maximizetheprobabilityofcoveringallthecells).
Yehoshua,AgmonandKaminka 5
2.The coverage percentage of the target area before the robot is first stopped (i.e., maximize the expected number of
coveredcells).
Let us now formally define these objective functions. First, we denote the coverage path followed by the robot by
A=(a ,a ,...,a ).Notethatm≥n,i.e.,thenumberofcellsinthecoveragepathmightbegreaterthanthenumberof
1 2 m
cellsinthetargetarea,sincetherobotisallowedtorepeatitssteps.WedefinetheeventS astheeventthattherobotis
A
notstoppedwhenitfollowsthepathA.Theprobabilitythattherobotisabletocompletethispathis:
(cid:89)
P(S )= (1−p ) (1)
A i
i∈(a1,...,am)
Thus,thefirstobjectiveistofindacoveragepathAthatmaximizestheprobabilityP(S ).Notethatinthisobjective,
A
theorderofvisitsofthecellsisnotimportant,aslongasthenumberofvisitsofthreatpointsalongthecoveragepathis
minimized(ideally,visitingeachthreatpointonlyonce).
Forthesecondobjective,wedenotethesequenceofnewlydiscoveredcellsalongthecoveragepathAby(b ,...,b ).
1 n
Notethatb (cid:54)=b foreachi(cid:54)=j,andthenumberofcellsinthissequenceisexactlythenumberofcellsinthegrid(n).For
i j
eachcellinthesequenceb ,wewilldenotethesub-pathinAthatleadsfromtheorigincella toitbyg .Letthenumberof
i 1 i
thenewcellsdiscoveredbytherobotuntilitisstoppedbeC .Then,underthethreatprobabilityfunctionp,theexpected
A
numberofnewcellsthattherobotvisitsuntilitisstoppedcanbeexpressedas:
(cid:88) (cid:89)
E(C )= (1−p ) (2)
A j
i∈(b1,...,bn)j∈gi
i.e.,E(C )isthesumoftheprobabilitiestoreachallthenewlydiscoveredcellsalongthecoveragepath.
A
Thus,thesecondobjectiveistofindacoveragepathAthatmaximizestheexpectedcoverageE(C ).Notethatinthis
A
objective,thevisitorderofthecellsiscrucial,sincetherobotistryingtocoverasmuchaspossiblebeforegettinghitbya
threat(ideally,coveringallthesafecellsbeforevisitingasinglethreatpoint).
Toillustratethesedefinitions,letusconsiderthefollowingsimplegrid,whichiscomposedof4cells:a ,a ,a and
11 12 21
a ,withtheprobabilitiesfordangerp specifiedineachcell.
22 ij
0 0.1
0.2 0.5
Assume that the initial location of the robot is in cell a (top left corner). Since there are no obstacles in this grid,
11
therearecoveragepathsthatvisiteachcellexactlyonce.Thesepathshavebothminimumlengthandmaximumprobability
tocomplete.Inourexampletherearetwosuchcoveragepaths:A = (a ,a ,a ,a )andA = (a ,a ,a ,a ).
1 11 12 22 21 2 11 21 22 12
Theirprobabilitytocompleteisthesameandequalsto:
P(S )=P(S )=1·0.9·0.8·0.5=0.36
A1 A2
However,thesepathsdon’thavethemaximumpossibleexpectedcoverage.Theexpectednumberofcoveredcellsin
A is:
1
E(C ) =1+1·0.9+1·0.9·0.5+1·0.9·0.5·0.8
A1
=1+0.9+0.45+0.36=2.71
AsimilarcomputationshowsthattheexpectednumberofcoveredcellsinA is:E(C )=2.56.
2 A1
6 TheInternationalJournalofRoboticsResearch0(0)
However, the path with the maximum expected coverage is A = (a ,a ,a ,a ,a ). To compute its expected
3 11 12 11 21 22
coverage,wefirstnotethatthesequenceofnewcellsdiscoveredalongthispathis(a ,a ,a ,a ).Thus,theexpected
11 12 21 22
coverageofA is:
3
E(C ) =1+1·0.9+1·0.9·1·0.8
A3
+1·0.9·1·0.8·0.5
=1+0.9+0.72+0.36=2.98
Therefore,bymakingoneadditionalstep,therobotisabletoraiseitsexpectednumberofcoveredcellsfrom2.71to
2.98.
In the following sections, we will use the notation S(A) to denote the survivability of the robot following coverage
pathA.S(A)mayrefertoeitherP(S )orE(C ),dependingonthecontext.
A A
Therobot’staskistoplanapaththroughT suchthateveryaccessiblefreecellinT (includingthedangerouscells)is
visitedbytherobotatleastonce.Inparticular,givenT,threequestionsmaybeasked:
1.WhatistheminimumcoveragetimeforT,andatwhatsurvivability?
2.WhatisthemaximumsurvivabilityforT,andatwhatcoveragetime?
3.Givenrequiredlevelsofsurvivabilityandcoveragetime,whatistheoptimalcoveragepath?
In order to help us answer these questions, we will define the following weighted cost function that takes both the
survivabilityandthecoveragetimefactorsintoconsideration.ForagivencoveragepathA,define
f(A)=−α·S(A)+β·|A| (3)
whereα,β ≥ 0aregivenupfront,accordingtotherequiredbalancebetweentheriskandthetimefactors.|A|isthe
numberofthestepstherobotneedstotakeinordertocompletethecoveragepath.1Thus,theproblemistofindacoverage
pathAthatminimizesthecostfunctionf(A),i.e.,f(A)≤f(B)forallpossiblecoveragepathsB.
Whenα=0,objective(3)translatestofindingaminimumtimecoveragepath,regardlessoftheriskinvolved.Achieving
thisobjectivewillprovideananswertoourfirstquestion,whichisequivalenttofindingasolutiontothegeneralcoverage
problem.Thismeansthatthecoverageproblembecomesaspecialcaseoftheadversarialcoverageproblem.Whenβ =0,
theobjectivetranslatestofindingacoveragepathwithaminimalrisk,withoutalimitonthepathlength.Achievingthis
objectivewillprovideananswertooursecondquestion.Lastly,settingfixedlevelsforαandβ willprovideananswerto
ourthirdquestion.Inthelastcase,theratioα/β willdeterminehowstronglytheobjectivepreferssafercoveragepaths
overshorterones.
3.1. ProblemComplexity
Wenowdiscussthecomplexityofthethreeproblemsofadversarialcoverage.Weanalyzethecomplexityofeachofthe
threeproblemsseparately.
Thefirstproblem,findingaminimumtimecoveragepath(α=0)isequivalenttotheproblemoffindingHamiltonian
walksingraphs.AHamiltonianwalkofagraphGisashortestclosedwalkthatpassesthrougheveryvertexofGatleast
once.TheproblemoffindingaHamiltonianwalkofagivengraphisknowntobeNP-Complete(Nishizekietal.,1983).
1Thetraveltimebetweencellsisuniformalongthegrid,thuscoveragetimeismeasureddirectlybythenumberofsteps.
Yehoshua,AgmonandKaminka 7
Wenowprovethatthesecondproblem,findingacoveragepathwithminimumrisk(β =0),isalsoNP-hard.Toprove
thatfindingthesafestcoveragepathishard,weneedtorefertothetwodifferentmethodsformeasuringthesurvivability
oftherobot(equations(1)and(2)).Weproveitseparatelyforeachmeasure.
Definition 3.1. The Safest Coverage with Minimal Risk Problem (SCMR): Given a grid representation of a world that
containsobstaclesandthreatpoints,findacoveragepathofthegridwithminimumtotalaccumulatedrisk.
Theorem1. TheSCMRproblemisNP-Hard.
Proof.Clearly,thisproblemisinNP,sinceonecaneasilyguessthecoveragepathoftherobotandthenverifyitsprobability
ofsurvivingitinpolynomialtime.ToproveitsNP-hardness,wewilluseareductionfromtheHamiltonianpathproblem
ongridgraphs.Agridgraphisafinitenode-inducedsubgraphoftheinfinitetwo-dimensionalintegergrid(seeFigure2
foranexampleofageneralgridgraph).TheHamiltonianpathproblemongridgraphs,i.e.,theconstructionofapaththat
visitseverynodeofthegridgraphpreciselyonce,hasbeenproventobeNP-completein(Itaietal.,1982).
Fig.2. Anexampleofageneralgridgraph.
GivenaninstanceoftheHamiltonianpathproblemonagridgraphG,weconstructaninstanceofthesafestcoverage
problemonatargetareaT asfollows.ForeachnodeinGwecreateacellwithathreatpointofprobabilitypinT.Allthe
othercellsinT willcontainobstacles.Thisconstructioncanbedoneinpolynomialtime.Clearly,thereexistsaHamiltonian
pathinG,ifandonlyifthesafestcoveragepath(withtheminimumaccumulatedrisk)inT visitseachthreatpointexactly
once.Therefore,wecanfindifthereexistsaHamiltonianpathinagivengridgraphG,byconstructingT,findingthesafest
coveragepathinT andthencheckingifitcoverseachthreatpointexactlyonce.Thus,SCMRisNP-hard.
Definition3.2. TheSafestPathwithMaximumExpectedCoverageProblem(SPMEC):Givenagridrepresentationofa
worldthatcontainsobstaclesandthreatpoints,findacoveragepaththatmaximizestheexpectedcoverageofthegrid.
Theorem2. TheSPMECproblemisNP-Hard.
Proof.The same construction from the previous proof can be used to prove the NP-Hardness of the SPMEC problem,
sincethetargetareaT thatwasconstructedtherecontainedonlythreatpointswithuniformprobabilitiesandobstacles(no
freecells).Thus,acoveragepathofT withmaximumexpectedcoveragepercentageisalsoacoveragepathwithminimum
accumulatedriskandviceversa.
Wenowprovethatthethirdproblem,findingacoveragepaththatmeetsrequiredlevelsofsurvivabilityandcoverage
time,isNP-Complete.
Definition3.3. TheGeneralAdversarialCoverageProblem(GACP):Givenagridrepresentationofaworldthatcontains
obstaclesandthreatpoints,doesthereexistacoveragepathwhoselengthis≤Landitsexpectednumberofcoveredcells
is≥α?
Theorem3. TheGACPproblemisNP-Complete.
Proof.SeeAppendixA.1.
Giventheprovenhardnessoftheproblem,itisnaturaltolookforapproximatesolutionalgorithms.
8 TheInternationalJournalofRoboticsResearch0(0)
4. STACAlgorithm
TheSpanningTreeAdversarialCoveragealgorithm(STAC,showninAlgorithm1)usesalayered-basedapproach.Itfirst
coversallthesafecells,usingaminimum-riskpathtomovebetweendisconnectedcells(ifthereareany).Thenitcovers
thedangerousareasfromtheleastdangerousonestothemostdangerousones,usingaminimum-riskpathtomovebetween
disconnected cells. This way the algorithm tries to cover as many safe cells as possible before attempting to cover any
dangerouscell,andthusmaximizethecoveragepercentagebeforetherobotisfirsthitbyathreat.
Algorithm1Spanning_Tree_Adversarial_Coverage(G,s)
Input:agridGandastartingcells
Output:acoveragepathP thatcoversallreachablecellsinGfroms
1: P ←∅
2: BuildthegraphH inducedfromthecellsinG
3: R←allcellsinGreachablefroms
4: GroupthecellsinRintol+1threatlevels,T0,...,Tl
5: foreverythreatleveli,0≤i≤ldo
6: Di ←{c|c∈Ti∧c(cid:54)∈P}
7: PDi ←Create_Safe_Coverage_Path(G,H,Di){Algorithm2}
8: if i>0then
9: fi ←lastcellinPDi−1
10: si ←firstcellinPDi
11: RunDijkstraonH fromfi
12: LetπibethesafestroutefromfitosifoundbyDijkstra
13: AddπitoP
14: AddPDi toP
15: returnP
STACbeginsbycreatingthegraphH = (V,E)inducedfromthegridcells(line2).Inthisgraph,eachcellinGis
representedbyavertexinV andverticesthatrepresentadjacentcellsinthegridareconnectedbyanedgeinE.STACruns
Depth-First-Search(DFS)onthisgraphinordertofindallthecellsthatarereachablefromthestartingcells(line3).Next,
itgroupsthereachablecellsintol+1(0≤l <n)distinctthreatlevelsaccordingtotheirassociatedthreatprobabilities,
startingfromlevel0whichcontainsallthesafecellsandendingwithlevellwhichcontainsthemostdangerouscells(line
4).
The loop in lines 5–14 iterates over all the threat levels from 0 to l. For each threat level, it creates a coverage
path that connects all the cells that belong to that level (lines 6–7). This is performed by using the procedure Cre-
ate_Safe_Coverage_Path(CSCP,showninAlgorithm2).Inordertocombineallthecoveragepathsforthedifferent
threatlevelsintoacompletecoveragepath,foreachtwoconsecutivethreatlevelsiandi+1,thealgorithmfindsthesafest
routebetweenthelastvisitedcellinleveliandthefirstvisitedcellinleveli+1.ThisisperformedbyrunningDijkstra’s
shortestpathsalgorithm(Dijkstra,1959)onthegraphH (lines11–12),usingthefollowingedgeweights:
(cid:40)
p /p ifcellj containsathreat
j min
w = (4)
ij
1/n otherwise
Thisweightfunctionensuresthatw ≥1foredgesthattargetadangerouscell,whilew =1/n(wherenisthegrid
ij ij
size)foredgesthattargetasafecell.Thisway,thecostofvisitingonedangerouscellisgreaterthanthecostofvisitingall
thesafecellsinthegrid.Thus,onlywhentherearetwoequallysafepathsconnectingtwonodes,wewillprefertheshorter
one.
Yehoshua,AgmonandKaminka 9
ThefinalcoveragepathP returnedbySTACconsistsofthecoveragepathsforallthethreatlevelsandtheroutethat
connectsthem.
TheprocedureCreateSafeCoveragePath(CSCP)findsthesafestcoveragepaththatconnectsallthecellsthatbelong
toaspecificthreatlevel.CSCPiscomposedoffivemainstages.
Algorithm2Create_Safe_Coverage_Path(G,H,C)
Input:agridG,itsinducedgraphH andagroupofreachablecellsC
Output:apathP thatcoversallthecellsinC
1: P ←∅
2: BuildthegraphGC inducedfromthegridcellsC
3: Findtheconnectedcomponents(areas)ofGC usingDFS
4: LetA1,...,Ak betheconnectedareasofGC
5: foreveryareaAi,1≤i≤kdo
6: RunSpiral-STConAi
7: LetπibethecoveragepathgeneratedbySpiral-STC
8: si ←firstnodeinπi
9: fi ←lastnodeinπi
10: foreverynodefi,1≤i≤kdo
11: RunDijkstraonthegraphH startingfromfi
12: foreverynodesj,j (cid:54)=ido
13: Findthesafestpathfi (cid:32)sj bytraversingtheshortestpathstree
14: BuildthegraphGAoftheconnectedareas
15: RunChristofidesonGA
16: LetA(cid:48),...,A(cid:48) betheorderoftheareasalongtheroutefoundbyChristofides
1 k
17: foreveryareaA(cid:48),1≤i≤kdo
i
18: Addthecoveragepathπ(cid:48) ofA(cid:48) toP
i i
19: if i<kthen
20: Addtheconnectingpathf(cid:48) (cid:32)s(cid:48) toP
i i+1
21: returnP
First,itfindsalltheconnectedareasofcellsthatbelongtothegiventhreatlevel(lines2–3).Aconnectedareaisdefined
asaconnectedsubsetofcellsinthegridthatbelongtothesamethreatlevel.Wealsodefinea safearea asaconnected
areacomposedofonlysafecellsanda dangerousareaasaconnectedareacomposedofonlydangerouscells.Figure3
showsanexampleofagridcontainingtwosafeareasandfivedangerousareasthatbelongtotwodifferentthreatlevels.
Thetwolow-threatlevelareasareoutlinedbyyellowlinesandthethreehigh-threatlevelareasareoutlinedbybluelines.
Fig.3. Anexampleofagridcontainingtwosafeareasandfivedangerousareasthatbelongtotwodifferentthreatlevels.Thelow-threat
levelareasareoutlinedbyyellowlinesandthehigh-threatlevelareasareoutlinedbybluelines.
10 TheInternationalJournalofRoboticsResearch0(0)
CSCPfindstheconnectedareasbyfirstbuildingthegraphG inducedfromitsinputgroupofcellsC (line2).This
C
graphisbuiltthesamewayastheinducedgraphH inAlgorithm1,butusingonlythecellsinC asverticesinsteadofthe
entiregrid’scells.ThenitrunsDepth-First-Search(DFS)onthisgraph(line3).
Inthesecondstage,CSCPfindsacoveragepathforeachconnectedareabyrunningtheSpiralSpanningTreeCoverage
(Spiral-STC)algorithm(GabrielyandRimon,2003)(lines5–9).Inthisalgorithm,asinglerobotisassumedtobeequipped
withasquare-shapedtoolofsizeD:Anypointwithinthetool’sareaistakentobevisited.Therobotmovesbyslidingthe
toolovertheareainanyofthefourbasicdirections(North,South,East,West).Theworkareaisapproximatelydivided
intoagridwithcellsofsizeD.Thegridisthenmadecoarse,suchthateachnewcellisofsize2D×2D.Cellswhich
arecompletelyoccupiedbyobstaclesarediscarded.Thecenter-pointsofalltheremainingcellsareconnectedtothoseof
their neighbors in the four basic directions, to form a graph. A spanning tree is then induced from the graph. The robot
followstheedgesofthisspanningtree,whilecoveringeach2D-cellinternally(seeexampleinFigure4).Ascanbeseen,
Spiral-STCgeneratesspiral-likecoveringpatterns.
Fig.4. Anexampleofaspanningtreecoverage.Blackcoloredcellsareoccupiedbyobstacles.Thespanningtree,denotedbyathick
blueline,connectsallthefreecoarsegridcells.Thecoveragepathisthedottedlinefollowingalongthespanningtree.
ThemainresultisthatSpiral-STCcoversanyplanargridinO(n)timeusingapathwhoselengthisatmost(n+m)D,
wherenisthenumberofD-sizecellsandm≤nisthenumberofboundarycellsdefinedbelow.Inpractice,m(cid:28)n,thus
Spiral-STCgeneratesclose-to-optimalcoveragepaths(GabrielyandRimon,2003).
Definition4.1. Boundarycellsarecellsthatshareeitherapointorasegmentwithacellcontaininganobstacle.
We have introduced a few changes to the original Spiral-STC algorithm. Specifically, the original algorithm always
returnstotheoriginalcellfromwhichitstartedthecoverage.Inourcasethereisnoneedtodothat.Assoonasallthe
cellsinagivenconnectedareaarecovered,therobotcanimmediatelyproceedtothenextarea.Inaddition,Spiral-STC
assumesthatthegivengridcontainsanevennumberofrowsandcolumns,thusitcanbedecomposedinto2D-sizecells.
Wehavehandledcoverageofareaswithanoddnumberofrowsand/orcolumns,byaddingonedummyrowand/orcolumn
ofobstaclestotheoriginalarea,beforerunningSpiral-STC.