外文翻译 空间机器人防碰路径规划文档格式.docx
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外文翻译 空间机器人防碰路径规划文档格式.docx
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SpaceRobotPathPlanning
forCollisionAvoidance
YuyaYanoshitaandShinichiTsuda
Abstract—Thispaperdealswithapathplanningofspacerobotwhichincludesacollisionavoidancealgorithm.Forthefuturespacerobotoperation,autonomousandself-containedpathplanningismandatorytocaptureatargetwithouttheaidofgroundstation.Especiallythecollisionavoidancewithtargetitselfmustbealwaysconsidered.Oncethelocation,shapeandgrasppointofthetargetareidentified,thosewillbeexpressedintheconfigurationspace.Andinthispaperapotentialmethod,Laplacepotentialfunction,isappliedtoobtainthepathintheconfigurationspaceinordertoavoidso-calleddeadlockphenomenon.Someimprovementonthegenerationofthepathhasbeenobservedbyapplyingpathsmoothingmethod,whichutilizesthesplinefunctioninterpolation.Thisreducesthecomputationalloadandgeneratesthesmoothpathofthespacerobot.Thevalidityofthisapproachisshownbyafewnumericalsimulations.
KeyWords—SpaceRobot,PathPlanning,CollisionAvoidance,PotentialFunction,SplineInterpolation
I.INTRODUCTION
Inthefuturespacedevelopment,thespacerobotanditsautonomywillbekeyfeaturesofthespacetechnology.Thespacerobotwillplayrolestoconstructspacestructuresandperforminspectionsandmaintenanceofspacecrafts.Theseoperationsareexpectedtobeperformedinanautonomousmannerinplaceofextravehicularactivitiesbyastronauts.
Intheabovespacerobotoperations,abasicandimportanttaskistocapturefreeflyingtargetsonorbitbytheroboticarm.Forthesafecapturingoperation,itwillberequiredtomovethearmfrominitialposturetofinalposturewithoutcollisionswiththetarget.
Theconfigurationspaceandartificialpotentialmethodsareoftenappliedtotheoperationplanningoftheusualrobot.Thisenablestherobotarmtoevadetheobstacleandtomovetowardthetarget.Khatibproposedamotionplanningmethod,inwhichbetweeneachlinkoftherobotandtheobstacletherepulsivepotentialisdefinedandbetweentheend-effecteroftherobotandthegoaltheattractivepotentialisdefinedandbysummingbothofthepotentialsandusingthegradientofthispotentialfieldthepathisgenerated.Thismethodisadvantageousbyitssimplicityandapplicabilityforreal-timeoperation.Howevertheremightbepointsatwhichtherepulsiveforceandtheattractiveforceareequalandthiswillleadtotheso-calleddeadlock.
Inordertoresolvetheaboveissue,afewmethodsareproposedwherethesolutionofLaplaceequationisutilized.Thismethodassuresthepotentialfieldswithoutthelocalminimum,i.e.,nodeadlock.InthismethodbynumericalcomputationLaplaceequationwillbesolvedandgeneratespotentialfield.Thepotentialfieldisdividedintosmallcellsandoneachnodethediscretevalueofthepotentialwillbespecified.
Inthispaperfortheeliminationoftheabovedefects,splineinterpolationtechniqueisproposed.Thenodalpointwhichisgivenasapointofpathwillbedefinedtobeapartofsmoothedsplinefunction.Andnumericalsimulationsareconductedforthepathplanningofthespacerobottocapturethetarget,inwhichthepotentialbysolvingtheLaplaceequationisappliedandgeneratesthesmoothandcontinuouspathbythesplineinterpolationfromtheinitialtothefinalposture.
II.ROBOTMODEL
ThemodelofspacerobotisillustratedinFig.1.
Therobotismountedonaspacecraftandhastworotaryjointswhichallowthein-planemotionoftheend-effecter.Inthiscasewehaveanadditionalfreedomofthespacecraftattitudeangleandthiswillbeconsideredtheadditionalrotaryjoint.Thismeansthatthespacerobotisthreelinkedwith3DOF(DegreeOfFreedom).Thelengthofeachlinkandtheangleofeachrotaryjointaregivenby
and
(i=1,2,3),respectively.Inordertosimplifythediscussionsafewassumptionsaremadeinthispaper:
-themotionofthespacerobotisin-plane,i.e.,twodimensionalone.
-effectofrobotarmmotiontothespacecraftattitudeisnegligible.
-robotmotionisgivenbytherelationofstaticgeometryandnotexplicitlydependingontime.
-thetargetsatelliteisinertiallystabilized.
Ingeneralin-planemotionandout-of-planemotionwillbeseparatelyperformed.Soweareabletoassumetheabovefirstonewithoutlossofgenerality.Thesecondassumptionderivesfromthecomparisonoftheratioofmassbetweentherobotarmandthespacecraftbody.Withrespecttothethirdassumptionwefocusongeneratingthepathplanningoftherobotandthisisbasicallygivenbythestaticnatureofgeometryrelationshipandisthereforenotdependingonthetimeexplicitly.Thelastonemeansthesatelliteiscooperative.
Fig.1ModelofTwo-linkSpaceRobot
III.PATHPLANNINGGALGORITHM
A.LaplacePotentialGuidance
ThesolutionoftheLaplaceequation
(1)iscalledaHarmonicpotentialfunction,anditsandminimumvaluestakeplaceonlyontheboundary.Intherobotpathgenerationtheboundarymeansobstacleandgoal.Thereforeinsidetheregionwherethepotentialisdefined,nolocalminimumtakesplaceexceptthegoal.Thiseliminatesthedeadlockphenomenonforpathgeneration.
(1)
TheLaplaceequationcanbesolvednumerically.WedefinetwodimensionalLaplaceequationasbelow:
(2)
AndthiswillbeconvertedintothedifferenceequationandthensolvedbyGauss-Seidelmethod.Inequation
(2)ifwetakethecentraldifferenceformulaforsecondderivatives,thefollowingequationwillbeobtained:
(3)
where
,
arethestep(cell)sizesbetweenadjacentnodesforeachx,ydirection.Ifthestepsizeisassumedequalandthefollowingnotationisused:
Thenequation(3)isexpressedinthefollowingmanner:
(4)
Andasaresult,twodimensionalLaplaceequationwillbeconvertedintotheequation(5)asbelow:
(5)
Inthesamemannerasinthethreedimensionalcase,thedifferenceequationforthethreedimensionalLaplaceequationwillbeeasilyobtainedbythefollowing:
(6)
InordertosolvetheaboveequationsweapplyGauss-Seidelmethodandhaveequationsasfollows:
(7)
where
isthecomputationalresultfromthe(n+1)-thiterativecalculationsofthepotential.
Intheabovecomputations,astheboundaryconditions,acertainpositivenumber
isdefinedfortheobstacleand0forthegoal.Andastheinitialconditionsthesamenumber
isalsogivenforallofthefreenodes.Bythisapproachduringiterativecomputationsthevalueoftheboundarynodeswillnotchangeandthevaluesonlyforfreenodeswillbevarying.Applyingthesamepotentialvaluesastheobstacleandinaccordancewiththeiterativecomputationalprocess,thesmallpotentialaroundthegoalwillbegraduallypropagatinglikesurroundingtheobstacle.Thepotentialfieldwillbebuiltbasedontheaboveprocedure.
Usingtheabovepotentialfieldfrom4nodalpointsadjacenttothenodeonwhichthespacerobotexists,thesmallestnodeisselectedforthepointtomoveto.Thisprocedurefinallyleadsthespacerobottothegoalwithoutcollision.
B.SplineInterpolation
Thepathgivenbytheaboveapproachdoesnotassurethesmoothlyconnectedone.Andifthegoalisnotgivenonthenodalpoint,wehavetopartitionthecellsintomuchmoresmallercells.Thiswillincreasethecomputationalloadandtime.
Inordertoeliminatetheabovedrawbacksweproposetheutilizationofsplineinterpolationtechnique.Byassigningthenodalpointsgivenbythesolutiontoviapointsonthepath,wetrytoobtainthesmoothlyconnectedpathwithaccurateinitialandfinalpoints.
InthispaperthecubicsplinewasappliedbyusingMATLABcommand.
C.ConfigurationSpace
WhenweapplytheLaplacepotential,thepathsearchisassuredonlyinthecasewheretherobotisexpressedtobeapointinthesearchingspace.Theconfigurationspace(C-Space),wheretherobotisexpressedasapoint,isusedforthepathsearch.ToconverttherealspaceintotheC-Spacethecalculationtojudgetheconditionofcollisionisperformedandifthecollisionexists,thecorrespondingpointintheC-spaceisregardedastheobstacle.Inthispaperwhenthepotentialfieldwasgenerated,theconditionsofallthepointsintherealspace,correspondingtoallthenodes,were
calculated.Thejudgmentofintersectionbetweenasegmentconstitutingtherobotarmandasegmentconstitutingtheobstacleateachnodewasmadeandiftheintersectiontakesplace,thisnodeistreatedastheobstacleintheC-Space.
IV.NUMERICALSIMULATIONS
Basedontheaboveapproachthepathplanningforcapturingatargetsatellitewasexaminedusingaspacerobotmodel.Inthispaperweassumethespacerobotwithtwodimensionaland2DOFroboticarmasshowninFig.1.
Thelengthofeachlinkisgivenasfollows:
l1=1.4[m],l2=2.0[m],l3=2.0[m],
andthetargetsatellitewasassumed1msquare.Thegrasphandle,0.1msquare,waslocatedatacenterofonesideofthetarget.Sothishandleisagoalofthepath.
Letusexplainthegeometricalrelationbetweenthespacerobotandthetargetsatellite.Whenweconsidertheoperationaftercapturingthetarget,itisdesirableforthespacerobottohavethelargemanipulability.Thereforeinthispapertheend-effecterwillreachthetargetwhenthemanipulabilityismaximized.Inthe3DOFcase,notdependingonthespacecraftbodyattitude,themanipulabilityismeasuredby
.Andifweassumetheend-effectorofthespacerobotshouldbeverticaltothetarget,thenallofthejointsanglesarepredeterminedasfollows:
Asallthejointsangles
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