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InternationalJournalofAutomationandComputing7

（2）,May2010,190-198

DOI:

10.1007/s11633-010-0190-8

EvolutionaryTrajectoryPlanningforanIndustrialRobot

1Introduction

Manyreal-worlddesigntasksinvolvecomplexmulti-

objectiveoptimizationproblemsofvariouscompetingde-

signspeci¯cationsandconstraints.Forsuchproblems,it

ishighlyimprobablethatallthecon°ictingcriteriawould

beextremizedbyasingledesign,andhence,atrade-o®

amongthecon°ictingdesignobjectivesisofteninevitable.

Inmathematics,multi-objectiveoptimisationseekstoopti-

miseavectorofnon-commensurableandoftencompeting

objectives,costfunctionsorperformancefunctionswithin

afeasibledecisionvariablespace.Intelligentrobotsystem

designisoneofthecomplexdesignproblems.

Theultimaterequirementofroboticsistocreateintel-

ligentroboticsystemsthatcanoperateautonomously.At

present,robotsareusedtoperformprogrammed,repeti-

tioustasksortaskswhereahumanoperatorhastocon-

stantlyspecifymotions.Incaseofautonomousrobots,the

robotisprovidedwithonlydescriptionsoftasksonanab-

stractlevelandwillcarryoutthosetaskswithouthuman

interventionorexplicitteaching.Inordertoreachthat

goal,moredevelopmentshouldtakeplaceintechnologies

ofperception,whichinvolvesautomatedreasoning,plan-

ning,manipulation,andlearning.Oneofthemainplan-

ningproblemsisthetrajectoryplanning,wheretheau-

tonomousrobothastoplanitsownmotions,and,byvirtue

ofthismotion,onlytherobotaccomplishesitstasks.The

classictrajectory-planningproblemisdescribedasfollows:

givenaninitialcon¯gurationanda¯nalcon¯gurationof

therobot,ithasto¯ndapathconnectingbothcon¯gura-

tionsthatavoidscollisionwithobstacles.Assumptionsare

thatthegeometryandthepositionofobstaclesareknown

inadvance,andobstaclesarestationary.

Inordertomaximizespeedofoperation,whicha®ects

theproductivityinindustrialsituations,itisnecessaryto

minimizethetotaltravellingtimeoftherobot.Therefore,

ManuscriptreceivedJune17,2008;revisedJuly10,2009

moreresearchworkshavebeencarriedouttogetminimum

timetrajectories[1¡4].

Therobottrajectoryplanningusingenergeticcriteria

providesseveraladvantages.Ityieldssmoothtrajectories

foreasiertrackingandreducesthestressestotheactuators

andtothemanipulatorstructure.Moreover,savingenergy

maybedesirableinseveralapplications,suchasthosewith

alimitedcapacityofenergysource（e.g.,robotsforspace

orunderwaterexploration）.Examplesofenergyoptimal

trajectoryplanningareprovidedinsomeliteratures.Both

optimaltravellingtimeandminimummechanicalenergyof

theactuatorsareconsideredtogetherasobjectivefunctions

insomeliteratures[5;6].

Fieldsofresearchsuchascomputergraphics,geomet-

ricdesign,androbotics（motionplanning）prefersmooth

trajectories,whichareachievedbyminimizingthejoints

jerk[7;8]andjointsacceleration[9].Gasparettoetal.[7;8]

consideredonlykinematicconstraints.Theydidnotcon-

siderdynamicconstraintssuchasjointtorques.Thecon-

ventionalmethod（numericaliterativeprocedure）usedby

ElnagerandHussein[9]cannotbeusedformulti-objective

problems.

Toobtainapracticaltrajectory（suchthattherobotdoes

notlooseanydegreeoffreedomatanystage）,themanipula-

bilitymeasurecanbeusedasthedecisioncriteriaforrobot

trajectoryplanning[10;11].Soinordertogetalltheabove

bene¯ts,alltheobjectivefunctionshavetobeconsideredin

acombinedmannertodooptimaltrajectoryplanning.But

noneofliteraturesconsideredalltheseobjectivefunctions

inacombinedmanner.

Manyauthorshavetreatedtheproblemoftrajectory

planningofrobotmanipulatorsinthepresenceof¯xed

obstacles[1;12].Whenmovingobstaclessharethesame

workspaceoccupiedbytherobotmanipulator,theopti-

misationofthetrajectoryde¯nedbytheend-e®ectoris

complex[1;5;6].Thiscomplexityisassociatedwiththelarge

numberofconstraintstobetakenintoaccountbytheop-

R.Saravananetal./EvolutionaryTrajectoryPlanningforanIndustrialRobot191

timiser.Theseconstraintsaretimedependentinthiscase.

Adesignmethodologyusingsequentialunconstrainedmin-

imisationtechniquesisproposedbySaramagoandJunior[5]

toobtaintheoptimalo®-linetrajectoryplanningofrobot

manipulators,whenoscillatingobstacleshavetobeavoided

bytheend-e®ector.Theirproblemofoptimaltrajec-

toryplanningconcernswiththedeterminationoftheend-

e®ectorrobotmotioninaminimumtimeandminimum

mechanicalenergybetweentwogivenpoints,whilesatisfy-

ingthelimitsoftheactuatore®ortsandavoidingcollision

withoscillatingand¯xedobstacles.

Themethodsthatareusedintheliteratures[1¡3;5;6;9¡14]

totacklethecomplexinstances（oscillatingobstaclesenvi-

ronment）havesomenotabledrawbacks:

1）theymayfailto

¯ndtheoptimalpath（orspendalotoftimeandmemory

storage）;and2）theyhavelimitedcapabilitieswhenhan-

dlingcaseswheretheconstraintsofmaximumacceleration

andmaximumdecelerationalongthesolutioncurveareno

longermet,orwheresingularpointsorcriticalpointsof

robotcon¯gurationexist.Toovercometheabovedraw-

backs,evolutionaryalgorithmscanbeused.Theadvan-

tagesofevolutionarytechniquesarefollows:

1）theyare

population-basedsearchalgorithms,soglobaloptimalsolu-

tionispossible;2）theydonotneedanyauxiliaryinforma-

tionlikegradients,derivatives,etc;3）theycansolvecom-

plexandmultimodalproblemsforglobaloptimality;and4）

theyareproblemindependent,i.e.,suitableforalltypesof

problems.Evolutionarytechniquesformulti-objectiveop-

timisationarecurrentlygainingsigni¯cantattentionfrom

researchersinvarious¯eldsduetotheire®ectivenessandro-

bustnessinsearchingforasetoftrade-o®solutions.Unlike

conventionalmethodsthataggregatemultipleattributesto

formacompositescalarobjectivefunction,evolutionary

algorithmswithmodi¯edreproductionschemesformulti-

objectiveoptimisationarecapableoftreatingeachobjective

componentseparatelyandleadthesearchindiscoveringthe

besttrade-o®solutions.

Themotivationsforusingpopulation-basedsearchtech-

niquesareasfollows:

1）The¯nalsolutionobtainedfromaconventionalmath-

ematicaloptimizationtechniqueisalwaysdependentonthe

input（i.e.,initialsolution）.Thewrongselectionofinitial

solutionmayleadtolocaloptimalsolution.Butincase

ofpopulation-basedtechniques,the¯nalsolutionwillnot

dependontheinitialsolution.So,the¯nalsolutionmay

beaglobaloptimalsolution.

2）Population-basedsearchtechniques（e.g.,evolution-

aryalgorithms（EAs））givemultidirectionalsearch.They

dealsimultaneouslywithasetofpossiblesolutions（theso-

calledpopulation）.Thisallowsusto¯ndseveralpossible

solutionsinasinglerunofthealgorithm,insteadofper-

formingaseriesofseparaterunsasinthecaseoftraditional

mathematicalprogrammingtechniques.

3）Optimizingalltheobjectivessimultaneouslyandgen-

eratingasetofalternativesolutionso®ersmore°exibility

todecisionmakers.Thesimultaneousoptimizationcan¯t

nicelywithpopulation-basedapproachessuchasEAsbe-

causetheygeneratemultiplesolutionsinasinglerun.

4）Whencomparedtoconventionalandmathematical

techniquesavailableinliterature[1¡12],evolutionaryalgo-

rithmsconvergequicklyandgivemorenumberofPareto

optimalsolutions.

5）Computationale±ciencyofevolutionaryalgorithms

isbetterthanthoseoftheconventionalandmathematical

techniquesavailableinliterature[1¡12].

6）Signi¯cantcomputationalspeed-upcouldbeachieved,

becausetheirrunningtimeisshorterthanthoseof

conventionalandmathematicaltechniquesavailablein

literature[1¡12].

Intelligentoptimizationalgorithmssuchasnon-

dominatedsortinggeneticalgorithm（NSGA-II）andmulti-

objectivedi®erentialevolution（MODE）areverymuch

desirablefortrajectoryplanningofanintelligentrealworld

robot.Trajectoryplanningforarealworldrobotisavery

complexandtedioustask,duetothefollowingreasons:

1）Theplanningalgorithmhastoconsiderthedynamic

modeloftherobot,whichisdependingontravellingtime,

payload,androbot0stask.Sotheplanningalgorithmisa

time-dependentone.

2）Inrobot0sworkspace,alltypesofobstacles（¯xed,

moving,andoscillatingobstacles）maybepresent.This

callsfortheplanningalgorithmtoconsideralltypesofob-

staclesforobstacleavoidance.Further,theinformation

abouttheobstaclesmaybepartiallyorfullyunknown.

Therefore,checkingforthepresenceofobstaclescollision

withrobotisaverycomplexandtimedependenttask.

3）Theenvironmentaroundtherobotisanever-changing

one.Thiscallsforplanningalgorithmtoupdatethedetails

fortrajectoryplanningforeachtimeinstant.

Thispaperconsidersallthedecisioncriteriafortheop-

timaltrajectoryplanningofindustrialrobotmanipulators

andtheobstacleavoidancecriteriaforoscillatingobstacles.

Inthispaper,twoevolutionaryalgorithms,namely,NSGA-

IIandMODE,areproposedtoobtainoptimaltrajectory

planningforanindustrialrobot（AdeptOneXLrobot）.

Twomethods,namelynormalizedweightingobjectivefunc-

tionsandaverage¯tnessfactor,areusedtoselectthebest

solutiontradeo®s.Twomulti-objectiveperformancemea-

sures,namelysolutionspreadmeasureandratioofnon-

dominatedindividuals,areusedtoevaluatetheParetoop-

timalfronts.Twomulti-objectiveperformancemeasures,