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AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume12|Number3|September2017|pp233–244 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2017.3.254 Originalscientificpaper

  

Determination of accuracy contour and optimization of workpiece positioning for robot milling 

Gotlih, J.a,*, Brezocnik, M.a, Balic, J.a, Karner, T. a, Razborsek, B.a, Gotlih, K.a aFaculty of mechanical engineering, University of Maribor, Maribor, Slovenia   

A B S T R A C T   A R T I C L E   I N F O

Workpiecepositioningintoamachine’sworkspacehasbecomeasimpletask.AdvancedCNCmachinesareequippedwith standardizedclamping systems,allowedworkpiecedimensionsarelistedinthemachine’sdocumentationandtolerancelevelsoftheendproducedpartsareknown.Thisgivesusersplentyof informationandgoodconfidencethattheyarechoosingthebestmachineforaspecifictask.Formoreuniversalmachines like industrialrobotsthis isnotthecase.DuetotheirflexibilityindustrialrobotscanbeanalternativetospecializedCNCmachines, butwhena specific task shouldbe executed, im‐portant information is missing. For a standard industrial robot themecha‐nismslayout,itsdimensionsanditsreachableworkspaceisknown,butaccu‐racy levels over the robot’s workspace are not. If a workpiece should bemilled within certain accuracy limits the robot’s documentation offers noinformationonhowcloseitcanbelocatedtothebordersoftherobot’swork‐space.Thisarticledealswiththementionedproblemwithanovelmethodolo‐gy. Based on experimental datawe found that a standard 6 DOF industrialrobot’sreachableworkspacecanbedividedintotworegions,onewithsuita‐blemillingaccuracyandanotherwithrapidlydecreasingmillingaccuracy.Toisolate the suitable accuracy region a regional non‐dominated sorting algo‐rithm was developed and an accuracy contour separating the regions wasextracted.Inthesecondpartofthearticleageneticsearchalgorithmbasedonregional non‐dominated sorting was applied to find the biggest arbitraryshapedworkpiece’ssize,positionandorientationinthesuitablemillingaccu‐racyregionoftherobot’sworkspace.

©2017PEI,UniversityofMaribor.Allrightsreserved.

  Keywords:RobotmillingAccuracycontoursWorkpiecepositioningNon‐dominatedsortingGeneticalgorithmOptimization

*Correspondingauthor:janez.gotlih@um.si(Gotlih,J.)

Articlehistory:Received26May2017Revised20July2017Accepted24July2017

  

1. Introduction 

Using industrial robots for less common tasks is becoming increasingly interesting among re‐searchers,asduetotheirflexibility,machiningcouldbecomeamoreregularapplicationforin‐dustrialrobots.ToreachthelevelwhererobotmachiningbecomesanacceptablealternativetoCNCmachining,robotmachiningaccuracyneedstobeanalysed,understoodandimproved.Themainreasonsforinaccuraciesinrobotmachiningwereclassifiedanddividedintothreegroupsby Kubela et al. [1] as environmental errors, robot dependent errors and process dependenterrors.Robotdependenterrorswerefurtherdividedintogeometrical,non‐geometricalandsys‐temerrors.Theirstudyonwoodmillingshowedthatthemostsignificanterrorsaretheresultofrobotsreversemotion,causingendproductshapeinaccuraciesinrangefrom0.3–0.5mm.Geo‐metricerrorcompensationhasbeenstudiedbyWuetal.[2],whoaddedafifthsteptotheusualrobot calibration approach, used to achieve the desired robot positioning accuracy. The pro‐posedsteps includemodelling,designofexperiments,measurement, identification,and imple‐

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mentation.Theirgoalwastoimprovethestandardapproach,sothataminimumnumberofex‐periments is required for robot’s calibration.With the proposed calibration technique an im‐provedpositioningaccuracywasachievedinabsoluterangeofupto0.17mm.Greaterposition‐ingaccuracywasmeasuredbyJózwiketal.[3],whousedahighspeedcameratoinvestigatetheeffectofdirectionof approachonpositioningaccuracy. In theirexperiment theauthors foundthatpositioninginaccuracyisraisingwithrobot’scontinuousoperation.AccuracyimprovementincontinuousmotionofaweldingrobotwasachievedbyMarôneketal.[4]byinsertionofaddi‐tionalpointsintotherobot’strajectory.Duetotechnologicalreasonsconstantspeedneedstobemaintainedduringwelding.Withhelpofexperimentstheauthorsfoundbettertrajectorymatch‐ingathighermovementspeeds.Jezeršeketal.[5]developedasystemforautomatictrajectorygenerationforroboticweldingbasedonrobotic laserscanning.Accuracy levelscomparabletotraditionalvisuallybasedtrajectorygenerationmethodswereachieved,butteachingtimeswereshortenedmorethan30‐times.Asystemforopticalaccuracyevaluationofrobotsinstaticanddynamicoperationmodewas setupbyPapaandTorkar [6]. In their experiment the authorsfoundthesamesubpixelaccuracyinbothmodes.

Topredict amechanism’s accuracy,differentdexterity relatedperformancemeasureshavebeen introduced in the past. Manipulating ability of end‐effector of robotic mechanisms wasstudiedbyYoshikawa[7]andamanipulabilityindexwasintroduced.Manipulability,definedasdeterminantoftheJacobianmatrixofaroboticmechanismisproposedasarobot’sperformanceindicator.OtherdexteritymeasuresbasedondeterminantoftheJacobianmatrixlikeconditionnumberandmonotonemanipulabilitywereproposed,assummarizedbyGotlihetal.[8].Intheirstudy authors demonstrated application of velocity anisotropy on a real industrial robot andintroduced a four dimensional robotic workspace for visualisation of robot’s manipulability.Problemsoflinkingperformancemeasurestoarobot’saccuracywereexposedbyMerlet[9].Inhisexperimentonlytherobot’smanipulabilityindexhasshownthesametrendasitsmeasuredaccuracy,butnoconnectionbetweenmanipulabilityandaccuracywasestablished.Toovercomeunitbasedinconsistency,transitionfromkinematictodynamicperformancemeasureswaspro‐posedandapowermanipulabilityindexwasintroducedbyMansouriandOuali[10].Thepowermanipulabilityindexisacombinationofkinematicandstaticparametersandincludestransla‐tionalandrotationalcomponents.Theauthorsshowedthatpowermanipulabilityisahomoge‐noustensorinsensitivetophysicalunitschange.

Optimizationofroboticsystemswithalgorithmsbasedonperformancemeasureswasstud‐iedbymanyauthors.Dynamicperformancemeasureswere found tobemoresignificant thankinematicbyVosniakosandMatsas[11].Theauthorsusedgeneticalgorithmforoptimalposi‐tioningofapredefinedmillingpathintoarobot’sworkspace.Theyfoundoutthatwithapplica‐tionofgeneticalgorithmanimprovementinrobot’smanipulabilityandareductioninjointtor‐quescanbeachieved.Geneticalgorithmwasalsousedforrobot’spathplanningtoavoidcolli‐sionofrobot’sstructurewithobstacles in itsworkspace[12].Anotherapproachofrobotpathplanning considering a mechanical power index and an obstacle avoidance index in a multi‐criteriaoptimizationproblemwassuggestedbyR.dosSantosetal.[13].Toovercomelocalmin‐imaauthorshaveusedatwo‐phaseoptimizationprocessandthesequentiallinearprogrammingmethod. Results showed good agreementwith Pareto optimality as better obstacle avoidancecausedanincreaseinpowerconsumption.Amulti‐criteriapathplanningalgorithmforroboticlaser cuttingwasdevelopedbyDolgui andPashkevich [14] and implemented in an industrialsoftware package, yielding significant reduction in process planning and changeover time forendusers.Athreestepoptimizationprocedureforworkpiecepositioning intoarobot’swork‐spacewasproposedbyLinetal.[15].Theauthorsdevelopedanapproachbasedonmanipulabil‐ityandstiffnessindexesandintroducedadeformationindextorefinethefinalworkspacemaps,wherecontourlinesdividingtherobot’sworkspaceintosubregionsindicateoptimalworkpiecepositioninglocations.

Topositionaworkpieceintoarobot’srefinedworkspaceprecisely,itisnecessarytoextractindividualcontourlines.Techniquesforcontourextractionaremainlyassociatedwithcomputervisionand shapedetectionandareused invariousengineering fields.Anapproach forwork‐piece localization and its shape deviations from nominal geometry in robotic deburring was

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Advances in Production Engineering & Management 12(3) 2017  235

proposedbyKussetal.[16].ApointcloudofaCADmodelwascomparedwithapointcloudrep‐resentingarealworkpieceinarobot’sworkspace.Thebestmatchingpointcloudswereusedtogenerateworkpiececontoursandtrajectoriesforthedeburringprocess.Intheirexperimenttheauthorsusedamillingrobotequippedwithastereocamera forshapesensingandachievedasignificantimprovementindeburringquality.Aroboticcellforopticalworkpiecedetectionwasdeveloped by Klancnik et al. [17]. The authors used neural networks for system calibrationwhere the relation between captured pictures and the robot’s coordinate systemwas learnt.Numeroustestsprovedthesystemadequateforuseinpractice.KarimiandNategh[18]intro‐ducedcontourmapstographicallyestimatekinematicerrorsinhexapodmachinesforanopti‐malworkpiecesetup.Theauthorsdevelopedananalyticalmodelforkinematicerrorestimationwith an average 11 % discrepancy from experimental results. Contour line generation fromshortlinesegmentswithageneticalgorithmbasedmethodwasproposedbyWeietal.[19].Anautomaticoctreebasedalgorithm for façadeandopeningcontourdetection frompointcloudsobtainedwithterrestriallaserscanningwasdevelopedbyTruong‐HongandLaefer[20]whereaknowledge databasewas used for contour validation. Relative errors ofmaximum 3%werefound between measured drawings and the reconstructed models. A method for buildingboundary identification based the Delaunay triangulation was developed by Awrangjeb andGuojun [21]. By defining amaximumpoint to point distance the algorithmproved capable ofdetectingconcaveedgesandholesofanypointcloudshape.

2. Material and methods 

Anexperimenttoevaluatetheeffectofdifferentperformancemeasuresonrobotmillingaccura‐cywas set upbyVeber [22] in hismaster’s thesis. A standard6DOF industrialmilling robotequippedwithamillingtoolandaclampingtablewasselectedfortheexperimentandthesys‐temwascalibrated.Toreduce influenceofhumanerrorseachexperimentwasrepeatedthreetimes.Workpieces of polymer compositematerialwith lowhardnesswere selected to reduceinaccuraciescausedbycutting forces.According to therobot’smanufacturer,workpiecehard‐nessshouldnotexceed35N/mm2anddeviationsinrangeof±0.12mmshouldbeexpecteddueto the robot’s rigidity.With all inaccuracy causes controlled,measured inaccuracieswere ex‐pectedtobeinfluencedmainlybytherobot’sdexterity.

Intheexperiment fivemillinglocationswereselectedandateachlocationthreework‐piecesweremilled in threesequentialruns.Thepositioningof theworkpieces ispresented inTable1,wherex,y,zrepresentdistancesinX,YandZdirectionfromtherobot’sbasecoordinatesystemanddistheresultingabsolutedistance.

Edgeandcircularpocketmillingoperationswereperformedoneachworkpiece.Endshapeswerethenmeasuredwithacoordinatemeasuringmachine.MeasurementresultscomparedtonominalvaluesarepresentedinTable2.

Table1Initialpositionsofthemeasuredworkpieces

Pos. x(mm) y (mm) z (mm) d(mm)1 1940 1053 610 22072 2060 ‐498 610 20603 1283 0 610 13764 1515 821 610 17235 1811 ‐709 610 1945

Table2Robotmillingmeasurementresults 

Pos.Experiment1 Experiment2 Experiment3

Nominalvalue(mm) Nominal value(mm) Nominalvalue(mm)20 80 20 80 20 80

1 20.134 80.122 20.136 80.122 20.134 80.1242 20.166 80.143 20.167 80.142 20.167 80.1433 20.477 80.522 20.476 80.523 20.476 80.5234 20.165 80.134 20.166 80.134 20.165 80.1365 20.155 80.134 20.155 80.134 20.155 80.134

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236  Advances in Production Engineering & Management 12(3) 2017

3. Results and discussion 

For result analysis absolute errors formeasured diameters and edge lengthswere calculatedandanormalizeddistanceparameter(NDS)expressedinpercentwasintroduced.NDSwascal‐culatedasthedistanceofaworkpiecefromtherobot’sbasecoordinatesystemdividedbythemaximumdistancetherobotcouldreachwithaverticaltoolorientation.

Fig. 1 reveals highest absolute errors atmilling position3, located close to 50%NDS. Forworkpieceslocatedat63%NDSandabove,absoluteerrorsarealmostconstantanddonotex‐ceed0.2mm,which isclose to the inaccuracycausedbytherobot’srigidity. Inrangebetween51% and 63%NDS,milling inaccuracy quickly raises. Due to highmeasurement consistencyindicatedby lowresultvariation,humanandcutting forcebased inaccuraciescanbeexcludedandthequickdecreaseofaccuracyatmillingposition3canbeattributedtotherobot’sdexterity.

Fig.2showsresultpointdeviationsfromanormaldistribution.Measurementresultsarerep‐resentedbysamplequantiles,whiletheoreticalquantilesrepresentastandardnormaldistribu‐tion.Forbothmeasuredquantitiesthehighestdeviationsarefoundathighestinaccuracies.Thisindicates a nonlinear dependency betweenworkpiece distance from the robot’s base and therobot’smilling accuracy, based onwhich a sharp boundary separating the robot’sworkspaceintotwomillingaccuracyregionsisexpected.

Fig.1Absoluteerrorsfordiameter(left)andedgelength(right)versusnormalizeddistance

Fig.2Absoluteerrordeviationsfordiameter(left)andforedgelength(right)

3.1 Accuracy contour for robot milling 

Inordertoextractthesuitablemillingaccuracyregionfromtherobot’sworkspace,aconnectionbetweenaccuracyanddexteritywasestablished.Becauseoftheunitbasedinconsistencyofma‐nipulability[10]wedidnot intendto findamodelconnectingaccuracywithdexterityover itscompleterange, insteadweonlyequatedaccuracyandmanipulabilityataconstantvalue.ThedexteritymeasuredefinedbyYoshikawa[7]wasfoundtobethemostreliableandwasthereforeused in our analysis.By assuming that samemanipulability causes samemilling accuracy,wewere able to extrapolate amilling accuracy boundary value to the robot’s three dimensionalworkspacetocreateaspatialaccuracycontour,separatingtheregionofsuitablemillingaccura‐cyfromtheregionofquicklyreducingmillingaccuracy.

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Asanexampleastandard6DOFrobot,describedbyDenavit‐Hartenberg(DH)notation,giveninTable3waschosen.

Withinversekinematicstherobot’sjointanglesanddexterityvaluesatallfivemillingposi‐tionswerefound.Toolorientationduringmillingwasparalleltotherobot’smainZaxis,there‐foreΘ4andΘ6 jointswereconsideredconstant.Additionally,allworkpieceswereprojectedtotheY=0plane, so that rotationabout robot’s first axisΘ1wasalso consideredconstant.AsaresultonlyΘ2,Θ3andΘ5contributedtodexteritycalculation.

Normalizeddexterity(ND)wasintroducedasactualdexteritydividedbythehighestdexteri‐tyofthemechanism.NDisexpressedinpercent(Eq.1).

max∙ 100 (1)

Table4showsrelativejointanglesandNDvaluesoftherobotatallfivemillingpositions.Ro‐bot’sinitialanglepositionswereΘ1=Θ2=Θ3=Θ6=0°,Θ4=180°andΘ5=90°.

ResultspresentedinTables2and4revealthattherapiddecreaseinaccuracyhappensbe‐tween22%and45%normalizeddexterity.Basedonourconclusionthatoneregionwithsuita‐blemillingaccuracyexistsandthatthetransitionbetweenthesuitableaccuracyregionandtherapidly decreasing accuracy region is sharp, 40 % normalized dexterity was chosen as theboundaryvalueforaccuracycontourgeneration.

Fig. 3 shows the clamping table coveredbynormalizeddexterity contours for vertical toolorientation.Workpiecesatactualmillingpositionsarerepresentedbyblacksquaresscaledby5.

Table3DHnotationofthediscussedrobot Link d(mm) a(mm) α (°) θ (°) Range(°)0 0 0 0 0 01 750 350 90 0 +/‐1852 0 1250 0 0 0/‐1463 0 ‐55 90 90 +155/‐1194 1100 0 90 180 +/‐3505 0 0 90 90 +/‐1256 0 0 0 0 +/‐350

Tooltip A B C D E

Table4JointanglesandNDatmillingpositions 

Pos. Θ1(°) Θ2(°) Θ3 (°) Θ4 (°) Θ5 (°) Θ6(°) ND (%)1 0 37.8 17.2 180 125 0 81.82 0 43.0 6.0 180 130 0 72.63 0 62.0 ‐35.7 180 153.8 0 22.24 0 53.0 ‐15.8 180 142.8 0 46.15 0 46.6 ‐1.8 180 135.2 0 63.9

Fig.3Topviewofworkpiecesonclampingtablewithdexteritycontours

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Togeneratetheaccuracycontour,regionsintherobot’sworkspacewithatlast40%normalizeddexteritywereisolated.Asrevolutionabouttherobot’sfirstaxisdoesnotaffectitsdexterityitwas possible to consider only the workspace cross section and revolve results to three‐dimensional space afterwards. According to this simplification a two‐ dimensional restrictedworkspace was created, only containing points with normalized dexterity values 40% andabove.Foramoregeneralsolutiontoolorientationwasconsideredfree.

Toisolatecontourpointsoftherestrictedworkspacearegionalnon‐dominatedsortingalgo‐rithmwasdeveloped.Thealgorithmsearchesfornon‐dominatedminimumpointsdefinedbyxandzintherestrictedworkspacepointcloud,whichrepresenttotherobot’stooltippositionsinX and Z direction relative to the robot’s base coordinate system. Extracted two‐ dimensionalcontourpointsformtheouterboundaryofthesubspaceintherobot’scompleteworkspace,in‐sidewhichmillingwithsuitableaccuracyispossible.PseudocodeofthealgorithmispresentedinFig.4. IC stands for iteration count,RC stands for region,LE stands fornumberof local ex‐tremesandQ1‐Q4areboolvariablesrepresentingquadrant1‐4.MAstandsformirroraxis,NnDstandsfornon‐dominatedandACstandsforaccuracycontour.Initializationvaluesarepresent‐edinTable5.

Tofindthefirstnon‐dominatedfrontintherestrictedworkspace,whichcorrespondstothefirst iteration inFig. 4, regularnon‐dominated search algorithmwas applied. In the followingiterations the restrictedworkspacewasmirrored over each axis to expose outer regions andisolatefrontpointscontributingtothefinalaccuracycontour.DuetotheshapeoftherestrictedworkspaceshowninFig.5,localextremepointsinXandZdirectionhadtobefoundinordertogenerateanequallydensecontour.Inourcaseonelocalextremepointwasdefinedininitializa‐tion dividing the concave region into two refined regions, but an arbitrary refinement levelwouldbepossibleifnecessary.Therefinedregionswererestrictedbysquaresrangingfromtheexisting regions contour’s end point to the enclosed local extreme point. The relevantmirroraxisforeachrefinedregionwasdeterminedbasedontheadjacentregion’smirroraxisandthetransformationTable6,whereQ1‐Q4representsquarequadrantsinXZplane(Fig.5)sequencedclockwisestartingatthetoprightcorner.

1 Initialization.2 IfIC<5:3 IfIC=1∨Q3=1:MA=None,NnDsort,Q3=0,IC+1,gotostep2.4 IfIC=2∨Q4=1:MA=X,NnDsort,Q4=0,IC+1,gotostep2.5 IfIC=3∨Q2=1:MA=Z,NnDsort,Q2=0,IC+1,gotostep2.6 IfIC=4∨Q1=1:MA=X∧MA=Z,NnDsort,Q1=0,IC+1,gotostep2.7 Else:8 IfRC=0:9 IfLE>0:RC=1,gotostep7.10 IfLE=0:Gotostep13.11 If0<RC<LE+1:RCboundaries.Findmirroraxes.Q(1‐4)=1,RC=RC+1,gotostep2.12 IfRC=LE+1:Gotostep13.13 JoinNnDregionsintosuitableaccuracycontour.

Fig.4 Pseudocodeforaccuracycontourgeneration

Table5Regionalnon‐dominatedsortingalgorithmvariablesandinitialvaluesRegionalNnD–variables Unit ValueIterationcount(IC) ‐ 1Regioncount(RC) ‐ 0Localextremes(LE) ‐ 1Quadrants(Q1‐Q4) ‐ 0

Table6TransformationtableforrefinedregiondefinitionQuadrant Changedirection Quadrant

Q1 Q2Q2 Q1Q3 Q4Q4 Q3

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Fig.5Crosssectionofrobot’scomplete(left)andrestricteddexterousworkspace(right)withprojectedworkpiecepositionsandaccuracycontour

Afterarefinedregionwasmirroredovertherelevantaxis,thenon‐dominatedsortingalgo‐rithmwasappliedtoextract theregionscontourpoints.Finally,allpartialresultswere joinedintotheaccuracycontour.

OntheleftsideofFig.5completedexterousworkspacecrosssectionwithaccuracycontourandfiveworkpiecepositionsisshown.Bluedotsrepresentsingularitypoints,whichareregionswithlowestdexterity,greenandyellowdotsrepresenttransitionregionswithmediumdexterityand red dots represent regions with highest dexterity. Accuracy contour is marked with adashedblackline.Workpiecesarepresentedintheprojectedplaneandscaledby5.Theplotontherightsideshowsthesuitablemillingaccuracyregionwithnormalizeddexterityvalues40%andabove, theaccuracycontourandall fiveprojectedworkpiecepositions.Foraccuracycon‐tourgenerationonlythebiggerpointcloudwasconsideredastherevolutionabouttherobot’smainZaxiscoverscomplete360°.

Now,thetoolorientationeffectbecomesvisible.DexteritycontourscorrespondingtoafixedtoolaxisshowninFig.3aredifferentfromdexteritycontoursshowninFig.5,wheretoolorien‐tationwasconsideredfree.Freetoolorientationenablesanotherdegreeoffreedom,whichin‐creasesthesizeofthesuitableregion.Byallowingfreetoolorientation,alsotheworkpiececlos‐esttorobot’sbaseislocatedinsidethesuitablemillingaccuracyregion,meaningthatallwork‐piecescouldbemilledwithsuitableaccuracy,regardingtheirsizeandassumingcorrecttoolandtechnologyselection.

3.2 Optimization of workpiece positioning 

Withtheseparationoftherobot’sworkspaceintotwomillingqualityregions,itisinterestingtofindthesize,positionandorientationofthebiggestworkpiecethatcanbemilledwithsuitableaccuracy.As the threedimensionalworkspaceof therobot inconsiderationcanbecreatedbyrevolving its cross sectionabout itsmainZaxis, it isobvious that thebiggestworkpiecewitheightcornerpointsisaparallelepiped. Geneticalgorithmwasappliedasthesearchtoolforthebiggestparallelepiped.Thealgorithmwassetupasanextensionoftheregionalnon‐dominatedsortingalgorithmusedtocreatetheaccuracycontour.PseudocodeofthealgorithmispresentedinFig.6whereICstandsforitera‐tioncount,MIformaximumnumberofiterations,PCforpopulationcount,PSforpopulationsizeandCCforconvergencecriteria.GAstandsforgeneticalgorithm,RPforreferencepointandNnDstandsfornon‐dominated.

Thealgorithmrequirescorrect initializationasshowninFig.6.Maximumallowediterationnumber, population size, reproduction, crossover andmutation frequency, mutation strengthandconvergencecriteriaweredeclaredinitializationvariables.Maximumnumberofiterationsdefinesthemaximumallowedlooprepetitions,populationsizedefinesthequantityofparallele‐pipedsgeneratedinoneiteration,reproductionfrequencydefinesthequantityofindividualstobetransferredtothenextgeneration,crossoverfrequencydefinesthefractionofindividualstoparticipateingeneexchangeandmutationfrequencydefinesthenumberofindividualstomu‐tatewithamagnitudeofgeometricalchangedefinedbymutationstrength.Convergencecriteriaisanearlyexitcriteria.ValuesusedforinitializationarepresentedinTable7.

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1 Initialization.2 IfIC=1:Boundariesforfirstgeneration,gotostep5.3 If1<IC<MI:Boundariesfornon‐firstgeneration,GAmechanics,gotostep5.4 IfIC=MI:Gotostep12.5 Generateparallelepipeds,addRPtorestrictedworkspacelist,NnDsort.6 IfRPisonfront:Discardparallelepiped,gotostep2.7 Else:Addparallelepipedtopopulation.8 IfPC<PS:Gotostep2.9 Else:10 IC=IC+1,10 IfCC=True:Gotostep12.11 Else:Gotostep2.12 Findglobalbest.

Fig.6Pseudocodeofthegeneticsearchalgorithm

Table7SearchalgorithmvariablesandtheirinitialvaluesGA‐variables Unit ValueMax.iterationnumber(MI) ‐ 100Populationsize(PS) ‐ 20Reproductionfrequency(RF) % 10Crossoverfrequency (CF) % 60Mutationfrequency(MF) % 30Mutationstrength(MS) ‐ 0.1Convergencecriteria (CC) ‐ 10

Togenerateaparallelepipedarandomcornerpointandtwoedgelengthsdescribingaparal‐lelogram in theY=0planewerecreated.The thirddimensionwasaddedbyextrusionof theparallelograminYdirection.Forthefirstiterationdifferentboundaryconditionstorestrictpar‐allelepiped’ssizescloselytotheallowedregionwereusedasforthefollowingiterations.Inthefirstiterationparallelepiped’ssizes,positionsandorientationsweregeneratedbyEqs.2‐8:

500, 500 (2)

0 (3)

500, 500 (4)

0, 1 ∙2

(5)

0, 1 ∙2

(6)

0, 1 ∙2

(7)

0, (8)

wherex,y,zarethebiggestparallelogram’scornerpointcoordinates,φisitsorientationabouttheYaxisandla,lbandlcaretheassociatedparallelepiped’sedgelengths.xmin,xmax,zminandzmaxaretheextremepointsoftheaccuracycontourinX,YandZdirection.Astheworkspaceisrota‐tionallysymmetricalxminandxmaxareusedasextremepointsinYdirectionandonlyoneorienta‐tionhastobeconsideredvariable.

After a parallelepiped was created and positioned in the robot’s workspace, its referencepointswereaddedtotherestrictedworkspacepointcloud,wherebycornerpointsofextrudedcornerswereprojectedtotheY=0plane.Becauseoftheconcaveshapeoftheaccuracycontouralsoedgepointshadtobecheckedbythecontourcriteria,thereforethreeequidistantreferencepointswerecreatedoneachedgeandaddedtotherestrictedworkspacepointcloud.Thenre‐gionalnon‐dominatedsortingwasappliedontheextendedpointcloud.Ifareferencepointwas

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foundnon‐dominated itwas locatedoutsideof theallowed spaceandaworkpiece containingsuchapointwould falloutsideof therobot’ssuitablemillingregion.Toavoidbad individualsproceeding to thenext generation, everyparallelepiped that failed theabove criteriawasdis‐cardedandanewonewascreatedinstead.Afterasmanyacceptableparallelepipedsasdefinedbypopulationsizewerecreated,theyweresortedbyrelevance,wherebiggervolumewascon‐sideredbetter.

ForthefollowingiterationsGAmechanismsselection,reproduction,crossoverandmutationwereappliedtocreatenewgenerationsofparallelepipeds.Forreproduction,onlythebestindi‐vidualsfromthelastgenerationwereused.Ifmultipleindividualsweretied,arandomselectionamongthemwasmade.Incrossovereachindividualfromthelastgenerationcouldparticipatemultipletimes.Toavoidreproductionanindividualcouldneverbecrossedwithitselfandatlastone genewas exchangedby theparents. Eqs. 9‐15wereused to generate new individuals bymutation, where index old defines the selected variables old value and MS defines mutationstrength.Eachindividualfromthelastgenerationcouldparticipateinmutationwithsameprob‐ability.

100, 100 ∙ ∙ (9)

0 (10)

100, 100 ∙ ∙ (11)

, 100, 100 ∙ ∙ (12)

, 100, 100 ∙ ∙ (13)

, 100, 100 ∙ ∙ (14)

0, ∙ (15)

Theoptimizationloopwasrepeateduntilaparallelepipedwasfoundtobethebestforsameiteration count as defined by convergence criteria or until maximum iteration number wasreached.On completion thealgorithmreturned theparallelepipedwith thebiggest volumeastheoptimizationtarget.Thebiggestparallelepipedrepresentsthesize,positionandorientationofthebiggestworkpiecethatcanbemilledwiththechosenaccuracywiththediscussedrobot. Inthefirstrunthealgorithmdidnotconvergewiththedefinedsettings.Workpiecesstillin‐creasedinsizefromgenerationtogeneration,indicatingspaceforimprovement.Forthisreasonanefficiencymechanismwasbuilt intothecode,generatingarestartfileifmaximumiterationnumberwasreachedbeforeconvergencecriteria.Fromtherestartfileanewoptimizationruncouldbestartedwith the individuals fromthe lastgenerationof thepreviousrunadopted fortheinitialgenerationofthenewoptimizationrun.Inourcase,sixoptimizationrunswereper‐formedtofindthetargetworkpiece.

LocationsoftwomostoppositecornerpointsofthebiggestworkpiecefoundbytheappliedgeneticalgorithmareAx=2000.23mm,Ay=‐1162.99mm,Az=1963.87mmandGx=1354.13mm,Gy=1162.99mm,Gz=292.81mm.Edgelengthsla,lbandlcarela=1002.44mm,lb=1486.24mmandlc=2325.98mm,wherelaisthedistancebetweentwocornerpointsonthesameedgeoftheparallelepipedinXdirectionandit isrotatedbyangleφabouttheYaxis incounterclockwisedirectioninfrontview,lbisthedistancebetweentwocornerpointsonthesameedgeofthepar‐allelepipedinZdirectionanditisrotatedbyangleφabouttheYaxisincounterclockwisedirec‐tioninfrontviewandlcisthedistancebetweentwocornerpointsonthesameedgeoftheparal‐lelepiped in Y direction.Workpiece rotation about the Y axis isφ = 3.37°. The volume of theworkpiece,calculatedastheproductofitsedgelengthsis3.47×109mm3.

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242  Advances in Production Engineering & Management 12(3) 2017

TheworkpieceispresentedinFig.7.Graphicsonthelefthandsideshowworkpiecesize,po‐sitionandorientationintherobot’scompleteworkspace,whilegraphicsontherighthandsideshow the sameworkpiece in the robot’sworkspacewithout singularities.Accuracy contour isplottedinY=0planeonly.Dexterityinblendcoloursisusedforworkspacepresentation,butforsize,positionandorientation searchonly theaccuracycontourmarked inblackwas relevant.Foreachcasefiguresarepresentedindefault,frontandintopview.

Fig.7 Optimizedworkpieceintherobot’sworkspacewithaccuracycontour:withsingularitypoints(left),withoutsingularitypoints(right)

4. Conclusion 

Thearticledemonstratesamethodology,basedoninnovativealgorithms,toisolatearefinedsubregioninarobot’scompleteworkspacethatallowsmillingwithsuitableaccuracy.Aparallelepi‐ped shaped, maximum sized workpiece and its position and orientation in a robot’s refinedworkspacewere foundasanexample,butanyarbitraryshapedobjectoranumberofobjectscanbepositionedintotherefinedregioninsteadbyusingtheproposedmethod.

Determination of accuracy contour and optimization of workpiece positioning for robot milling

The presented methodology can also be applied to robotic mechanisms in different manufac-turing environments. Instead of defining the accuracy contour by milling measurements, other machining processes to create performance contours can be used. Contours for a robot’s posture accuracy for grasping and positioning, measurement accuracy contours for tolerance control of big welded parts and many more can be created to optimally restrict a robot’s workspace. Some analytical performance measures that can be used for contour generation were also discussed in this article.

It is to be noted that complex contour shapes may require a different extraction approach than the regional non-dominated sorting algorithm presented in this article. The algorithm could be used because of simplicity of the accuracy contour, containing only two inflection points that were easy to identify and was tailored for quick contour extraction as it was intended to serve as the optimization function for the search algorithm. A general contour extraction algorithm would consider each point from the initial point cloud more often and would be less suitable for workpiece size and placement optimization.

Genetic algorithm was applied for workpiece size, position and orientation search, because of good results reported for similar problems by other authors. Discard penalty was introduced for individuals violating the accuracy contour criteria. Because of the penalty each new iteration took longer to complete as workpieces from earlier generations were closer to contour bounda-ries and discarded more often. An algorithm with faster convergence is being developed, allow-ing a more general contour generation algorithm to be used as optimization function.

Acknowledgement The authors would like to thank the Ministry of Higher Education, Science and Technology of the Republic of Slovenia, the University of Maribor, Slovenia and the Faculty of Mechanical Engineering of Maribor, Slovenia, who made the presented study possible by financial and infrastructural support.

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