王林：車間運(yùn)輸機(jī)器人系統(tǒng)設(shè)計(jì) word文檔可自由復(fù)制編輯word文檔可自由復(fù)制編輯0FsK(6)其中上標(biāo)r和f分別代表有效自由度和實(shí)際自由度。K為對(duì)角矩陣，其對(duì)角線上的子矩陣是減少了的有效矩陣B以連桿變量的形式出現(xiàn)的。為了驗(yàn)證子矩陣在方程（4，5）k中是否正確，ki和ki可表示如下：pqkipqkpqrskirskip,r=1,2,3;q=1,…,nr;s=1,…,12(7a)pqkikpqkipqrsrskip,r=1,2,3;q=1,…,m;s=1,…12(7b)其中ki是元件形狀函數(shù)，nr是連桿變量數(shù)，m是模塊變量數(shù)。方程式中的標(biāo)注即多次出現(xiàn)的下標(biāo)指數(shù)是以概括的形式出現(xiàn)的，這些下標(biāo)只不過(guò)是公式的一部分，并不表示某一含義除非特定指明。這些子矩陣可表示成：MffNEk[mkikk(kkikki)PkikikiRki]（8a） qt pq pt pq ptzs ptpqzs zs pqzsptuvzusvk1i1NEk Mrf [mkikk(kkikki)PkikikiRki]（8b） qt pqpt pq ptzs ptzspqzs pzusvqzs ptuvk1i1其中PkikikidV和RkikikikidV；z,u=1,2,3；uv Vki uv zusv Vki zsuvmkis,v=1,?,12是時(shí)間變量， 是第k個(gè)機(jī)構(gòu)的第i個(gè)元件的質(zhì)量。在定義 . . .k. .k. .ki . .ki .ak和bki時(shí)，柯氏力和地心引力可由下列算式計(jì)算出m mq q mp p mru mqru q mpru p來(lái)：NEk Qr [mkikak(kbkiakki)PkikibkiRki]（9a） q pqp pqpzs ppqzs zs pqzspuvzusvk1i1QfNEk[mkikak(kbkiakki)PkikibkiRki]（9b） q pqp pqpzs p pqzs zs pqzspuvzusvk1i1這個(gè)運(yùn)動(dòng)方程式綜合了變量步長(zhǎng)和變量預(yù)測(cè)校正的算法，以獲取坐標(biāo)系和i i中的時(shí)間記錄。于是，有關(guān)物體參考系的節(jié)點(diǎn)位移可由模塊轉(zhuǎn)換公式獲得。由應(yīng)力i與位移關(guān)系式計(jì)算出零件受到的壓應(yīng)力。整個(gè)參考系中各點(diǎn)的位移可用和機(jī)架的各節(jié)i點(diǎn)位移算出。點(diǎn)的偏移可由那個(gè)點(diǎn)在實(shí)際運(yùn)動(dòng)和有效運(yùn)動(dòng)的位移差精確的求出。3減少約束對(duì)機(jī)器手進(jìn)行動(dòng)態(tài)分析的方法就是計(jì)算N個(gè)獨(dú)立點(diǎn)在同一時(shí)間內(nèi)的運(yùn)動(dòng)。因此，約束t數(shù)目最好滿足NN，而且這么多的約束在優(yōu)化設(shè)計(jì)時(shí)也是不切實(shí)際的。不過(guò)有一個(gè)很 c t有效的辦法可以使約束數(shù)控制在N范圍內(nèi)又可以使約束數(shù)滿足t的所有值，這就是用cKreisselmeier-Steinhauser函數(shù)[3等量替換單個(gè)時(shí)間約束，此函數(shù)表示如下：] g(x)1lnNtexp(cg)j c jnn1其中g(shù)(X)g(x,t)和C是正數(shù)并由g和gjn之間的關(guān)系決定即min(gjn).jn j n j這可以說(shuō)明Kreisselmeier-Steinhauser函數(shù)限定了一個(gè)保守的值域[4]比如g總是比jmin(gjn)更重要，而且c的值越大g和min(gjn)之間的差就越小。這就是所謂用最j關(guān)鍵的約束等量替換了諸如g(x)min[g(X)]j jn在這一方法中，用等量約束g限定了分段函數(shù)并使其由g向g間斷的過(guò)渡。在這 j jp jq一值域里盡管左右突出的構(gòu)件在過(guò)渡點(diǎn)有差異，但他們具有相同的標(biāo)識(shí)和梯度，因此可在過(guò)渡點(diǎn)自然結(jié)合。隨著時(shí)間逐步的趨近零點(diǎn)，等量約束也變得逐漸光滑。上述所提到的非線性約束優(yōu)化問(wèn)題可以由NLPQL[11]來(lái)解決，即運(yùn)用序列二次方程的方法。這種優(yōu)化需要初始信息df/dx和dg/dx，m=1,?,N這兩個(gè)可由目前研究出的mjm V有限差來(lái)計(jì)算。4舉例雙桿平面機(jī)器人如圖1所示。運(yùn)動(dòng)原理是被動(dòng)塊E沿直線從初始位置（θ1=120，°θ2=-150°）運(yùn)動(dòng)到終點(diǎn)位置（θ1=60，°θ2=-30°）。E的運(yùn)動(dòng)軌跡表示如下： 0.5 T 2t△X△Y(tsin)E E T 2T整個(gè)運(yùn)動(dòng)過(guò)程的時(shí)間T=0.5s。每一個(gè)連桿的長(zhǎng)度為0.6米并由兩個(gè)等長(zhǎng)的零件連接著。其零件的外徑D，其為本設(shè)ki計(jì)的變量，k=1,2；i=1,2。零件的厚度為0.1D。物體的壓強(qiáng)和密度分別是kiE=72GPa,ρ=2700Kg/m-3。模塊變量縮小了形狀尺寸。最先結(jié)合的兩個(gè)模塊和最先有著固定自由的約束條件的軸也都被考慮到了。位于連接點(diǎn)B處的桿2質(zhì)量為2kg，被動(dòng)物塊和有效載荷的總質(zhì)量為1kg。設(shè)計(jì)的約束條件如下：n-75MPa≤σi≤75MPai=1,…,sδ≤0,001mn其中應(yīng)力約束由節(jié)點(diǎn)頂部或底部的s個(gè)點(diǎn)來(lái)驗(yàn)證。δ是E的實(shí)際運(yùn)動(dòng)軌跡與有效運(yùn)動(dòng)軌跡的偏離量（即x和y方向的最大偏移值）。初始設(shè)計(jì)變量D均為50mm。ki圖1平面機(jī)器手操作器在這個(gè)例子里，等量約束是由最關(guān)鍵的約束組成的并且其結(jié)果與Kreisselmeier-Steunhauser函數(shù)的結(jié)果進(jìn)行了比較。后者函數(shù)中適用了c的不同值，可以發(fā)現(xiàn)c的值越小其產(chǎn)生的設(shè)計(jì)就越死板。c=50時(shí)的設(shè)計(jì)是最理想的。應(yīng)當(dāng)指出的是編譯器的限制可能會(huì)超過(guò)c的最大值，這完全取決于指數(shù)函數(shù)也就是只要設(shè)計(jì)變量的低限足夠的小。另一方面，最關(guān)鍵的約束會(huì)產(chǎn)生極小質(zhì)量的設(shè)計(jì)并且精確的迎合偏移位移量。最小的質(zhì)量，恰當(dāng)?shù)闹睆胶头磸?fù)運(yùn)動(dòng)的次數(shù)在表4-1中列出。設(shè)計(jì)軌跡見(jiàn)表4-2。表KS-c表明了由Kreisselmeier-Steinhauser函數(shù)產(chǎn)生的結(jié)果，然而MCC表示關(guān)鍵約束?？梢?jiàn)應(yīng)力遠(yuǎn)遠(yuǎn)小于允許值，因此應(yīng)力約束受到了限制。連桿2中間的應(yīng)力最大見(jiàn)圖4-3。被動(dòng)物塊的偏移量δ的最佳解決方案見(jiàn)圖4-4圖2設(shè)計(jì)參數(shù)表1平面機(jī)器人控制器最佳方法圖圖3頂部連接兩個(gè)的平均壓力的最佳設(shè)計(jì)圖4最終效應(yīng)器偏差的最佳設(shè)計(jì)5總結(jié)在研究中，高速遙控操縱器的最佳設(shè)計(jì)方案取決于動(dòng)態(tài)特性。操縱器的固定軌跡與實(shí)際軌跡運(yùn)動(dòng)也必須考慮到。把最關(guān)鍵的約束用作等量約束。最關(guān)鍵的約束的時(shí)間點(diǎn)可能隨著設(shè)計(jì)變量的改變而變化。這表明分段的等量約束并不會(huì)使設(shè)計(jì)過(guò)程產(chǎn)生缺陷。序列二次方程用于解決設(shè)計(jì)問(wèn)題，其是運(yùn)用整體偏差進(jìn)行靈敏度計(jì)算。高速平面遙控操縱器已被優(yōu)化設(shè)計(jì)成在應(yīng)力和偏差限制下的最小質(zhì)量。基于Kreisselmeier-Steinhauser函數(shù)產(chǎn)生的保守設(shè)計(jì)下使用等量約束，最好的設(shè)計(jì)理念就是用最關(guān)鍵的約束。附錄附錄BHigh-speedflexiblemanipulatoroptimizationdesignAbstract-Amethodologyispresentedfortheoptimumdesignofroboticarmsundertime-dependentstressanddisplacementconstraintsbyusingmathematicalprogramming.Felementsareusedinthemodelingo1"theflexiblelinks.Thedesignvariablesarethecross-sectionaldimensionsoftheelements.Thetimedependenceoftheconstraintsisthroughtheuseofequivalentconstraintsbasedonthemostcriticalconstraints.ItithisapproachyieldsabetterdesignthanusingequivalentconstraintsobtainedbytheKresselmeier-Steinhauserfunction.Anoptimizerbasedonsequentialquadraticprogrammiusedandthedesignsensitivitiesareevaluatedbyoverallfinitedifferences.Thedynamicalequationscontainthenonlinearinteractionsbetweentherigidandelasticdegrees-of-frillustratetheprocedure,aplanarroboticarmisoptimizedforaparticulardeploymentusingdifferentequivalent1997ElsevierScienceLtd.1.INTRODUCTIONTheincreasingdemandforhigh-speedrobotshasmadeitnecessarytousecomponentsmustbedesignedforminimumweight.Thetraditionaldesignofroboticarmsbasedonmuposturesinstaticregimeisnotsuitableforhigh-speedsystemswherethestressesandaregovernedbythedynamiceffects.Topreventfailure,intricateinteractionsbetweeandelasticmotionsmustbetakenintoaccountinthedesign.Thedesignofstructuralsystemsundertransientloadinghasbeenstudiedbyusingequivalentconstraintsbasedoncriticalpointselection[1],timeintegralofviolatedandKreisselmeier-Steinhauserfunction[3,4].Incriticalpointselection,itisassulocationofthecriticalpointsareassumedtobefixedintime,howeverthisassumptionisnotappropriateforhigh-speedmultibodysystem's.Thesecondapproachhasthedisadvantagtheequivalentconstraintiszerointhefeasibledomainandhencethereisnoindicaticonstraintisalmostcritical.TheuseofKreisselmeier-Steinhauserfunctionresultsinanequivalentconstraintwhichisnonzerointhefeasibledomain,howeveritdefinesaconenvelopeandyieldsoversafedesigns.Inthedesignofroboticarms,theconventionalapproachistoconsidermultiplestaticpostures[5-7]ratherthanconsideringthetime-dependencyoftheconstraints.Thisapnotappropriateforhigh-speedsystems,sinceafewposturescannotrepresenttheoveramotion,andfurthermthedisplacementsandstressescomputedareinaccurateduetoomittcouplingbetweenrigidandelasticmotions.Infact,thiscouplingistheessenceofaflexiblemultibodyanalysis[8-10].Inthisstudy,amethodologyforthedesignofhigh-speedroboticarmsisdevelopedconsideringthecoupledrigid-elasticmotionofthesystemandthetime-dependencyoftheconstraints.Themostcriticalconstraintsareusedastheequivalentconstraints.Theofthemostcriticalconstraintsmayvaryasthedesignvariableschange.Thesensitiresponsei$evaluatedbyoverallfinitedifferencesandtheoptimizationiscarriedoutbysequentialquadraticprogramming[11].Toillustratetheprocedure,atwo-linkplanarrobisoptimizedforstrengthandrigidity.TheresultsarecomparedwiththoseobtainedbyKreisselmeier-Steinhauserfunction.2.DESIGNPROBLEMInthissection,theoptimumdesignisformulatedasanonlinearofaroboticmathprogrammingproblemforstrengthandrigidity.ThearmconsistsofNnumberofflexibleachofwhicharediscretizedbyEknumberofbeamfiniteelements.Theobjectiveistominimizetheweightofthearm.Thecontraintsrelatedtostrengtharetheelementstrtheconstraintsforrigidityarethedeviationsoftheselectedpointsfromthepathoftherigidmodel.Thedesignvariablesarethecross-sectionalpropertiesofthelinkelements.Mathemthisiswrittenasminimizetheobjectivefunctionsubjecttoconstraints.fNEkkiVkik1i1g(x,t)0j1,..N.,（1）j cwherekiandVkiarethemassdensityandvolumeoftheithelementofkthbody,respectively,xisthevectorofNnumberofdesignvariablesandNisthetotalnumberof V Ctime-dependentconstraints.Inevaluatingthedisplacementsandstresses,thefollowingformulationbasedonRef.[10]isemployedtomodelthecoupledrigid-elasticmotionofLetthedeformationofalinkBbedefinedrelativetoalinkreferenceframekwhichkfollowstheglobalmotionofBinamannerconsistentwiththeboundaryconditions.Theknumberofelasticdegrees-of-freedomofeachlinkisreducedbymodalreduction.Thegeneralizedcoordinatesofthesystemarethejointvariablesandmodalvariables. i iThevelocityofaparticleP,ki,canbewrittenasvkikiki(2)wherekiandkiarethecorrespondinginfluencecoefficientmatrices.Kaneetal.'sequations[12]areusedtodeterminetheequationsofmotionas M QFsF(3)ywherey[T,T]Tisthevectorofgeneralizedspeeds,Fisthevectorofgeneralizedappliedforces,andM,QandFsarethegeneralizedmasses,Coriolisandcentrifugalforcesaelasticforces,respectively,asshownbelow:MrrMrf M NEkkikiTkikikidV(4) symMff V k1i1 kisymkiTkiQrNEkkiTQki(kiki)dV（5） Qfk1i1 VkikiT0FsK(6)wherethesuperscriptsrandfrefertorigidbodyandelasticdegrees-of-freedom,respectively.KisablockdiagonalmatrixwhosediagonalsubmatricesarethereducedmatricesofBintermsofmodalvariables.Toevaluatethesubmatricesineqkiands(4,5),kikareexpressedinthefollowingformas:pqkipqkpqrskirskip,r=1,2,3;q=1,…,nr;s=1,…,12(7a)kikkikip,r=1,2,3;q=1,…,m;s=1,…12(7b)pq pq pqrs rswherekiistheelementshapefunction,nristhenumberofjointvariablesandmnumberofmodalvariables.Notethatintheequations,arepeatedsubscriptindexinatermimpliessum-mation.Superscriptsaregenerallypartofthelabelinganddonotimplysummationunlessotherwisespecified.ThemasssubmatricescanbewrittenasMffNEk[mkikk(kkikki)PkikikiRki]（8a） qt pq pt pq ptzs ptpqzs zs pqzsptuvzusvk1i1MrfNEk[mkikk(kkikki)PkikikiRki]（8b） qt pq pt pq ptzs ptpqzs zs pqzsptuvzusvk1i1 wher PkikikidV and RkikikikidV uv Vki uv zusv Vki zsuvz,u=1,2,3;s,v=1,?,12arethetime-invariantmatrices-andmk'isthemassofithfiniteelementofthekth . ..k..k..ki..ki.body.Bydefiningakandbki,theCoriolisand m mq q mp p mru mqru q mpru pcentrifugalforcescanbecomputedasQrNEk[mkikak(kbkiakki)PkikibkiRki](9a) q pqp pqpzs ppqzs zs pqzspuvzusvk1i1QfNEk[mkikak(kbkiakki)PkikibkiRki](9b) q pqp pqpzs p pqzs zs pqzspuvzusvk1i1Theequationsofmotionareintegratedbyusingavariablestep,variableorderorrectoralgorithmtoobtainthetimehistoryofthegeneralizedcoordinatesand.Theninodaldisplacementswithrespecttothebodyreferenceframesareobtainedbythemodaltransformationof.Theelementstressesarecomputedbythestress-displacementrelationsidisplacementsofthepointsofinterestintheglobalreferenceframearefoundbyusiandithenodaldisplacementsinthebodyframes.ThedeviationofapointisdefinedasthebetweentheglobaldisplacementsofthatpointintheHexibleandrigidmodels.Itshouldbenotedthat,intheequationsofmotion,theonlytermsthatarefunctionsofdesignvariablesarethestiffnessmatrix,theelementmassesandthearraysPkiandRkiinthemassmatrixandloadvector.Henceintheanalyticalsensitivityanalysis,thesearethshouldbedifferentiatedwithrespecttothedesignvariables.However,analyticalevathesensitivitisesadifficulttaskinthisclassofproblems.Asemi-analyticaloroverallfinitedifferenceapproachismuchbettersuited.3.COIVSTRAINTREDUCTIONThedynamicresponseofthearmiscalculatedatN,numberofdiscretepointsinthetimetdomain.Hence,thenumberofconstraintstobesatisfiedbecomesNN，andsuchac tlargenumberofconstraintsisnotpracticalinanoptimizationprocess.AneffectiveapproachtokeepthenumberofconstraintsasNandtoensuresatisfactionofcconstraintsforallvaluesoftistodefineequivalenttime-independentconstraintsbyusingKreisselmeier-Steinhauserfunction[3]asg(x)1lnNtexp(cg)j c jnn1whereg(X)g(x,t)andcisauser-selectedpositivenumberwhich jn j ndeterminestherelationbetweengandthemostcriticalgjn,min(gjn).ItcanbeshownjthattheKreisselmeier-Steinhauserfunctiondefinesaconservativeenvelope(4]suchthgisjalwaysmorecriticalthanmin(g),andthelargerthevalueofc,theclosergfollows jn jmin(g).Thissuggestsusingthemostcriticalconstraintastheequivalentconstraintasjng(x)min[g(X)]j jnInthisapproach,theequivalentconstraintgdefinesapiecewise-smoothfunctionwithjfinitediscontinuousgradientsasitmakestransitionsfromgtog.Inthisenvelope, jp jqalthoughtheright-andleft-handderivativesaredifferentatthetransitionpoints,tsamesignandthegradientsareblendedatthetransitionpointsbythenumericaldiffeInthelimitasthetimestepapproacheszero,theequivalentconstraintbecomessmooThenonlinear,constrainedoptimizationproblemdefinedaboveissolvedbyusingtheoptimizerNLPQL[1l]whichisbasedonsequentialquadraticprogramming.Thisoptimizerrequiresfirst-orderinformationdf/dxanddg/dx,m=1，2，N;whicharecomputedbyoverall m j m Vfinitedifferencesinthepresentwork.4.NUMERICALEXAMPLEAtwo-linkplanarrobotisshowninFig.1.Asingletaskisconsideredinwhichtheend-effectorEisrequiredtodeployfromaninitialposition（θ1=120，°θ2=-150°）toafinalposition（θ1=60，°θ2=-30°）alongastraightline.TheprescribedmotionofEisgivenas 0.5 T 2t△X△Y(tsin) E E T 2TTheperiodofthedeploymentmotion,T,istakentobe0.5s.Eachlinkisoflength0.6mandismodeledbytwoequallengthtubularEulerbeamelements.Theouterdiameters,D,k=1,2;i=1,2oftheelementsaretakenasthedesignkivariables.Thewallthicknessofeachelementissettobe0.1D.ThematerialpropertiesarekiE=72GPaandp=2700kg/m-3.Theproblemsizeisreducedbyusingmodalvariables.Thetwobendingmodesandthefirstaxialmodewithfixed-freeboundaryconditionsareconsTheactuatoroflink-2islocatedatjoint-Bhasamassof2kgandthecombinedmassoftheend-effectorandpayloadis1kg.Thedesignproblemissolvedunderthefollowingconstraints:n-75MPa≤σi≤75MPai=1,…,sδ≤0,001mwherethestressconstraintsareevaluatedatnnumberofpointswhicharethetopandsbottompointsateachnode.δisthedeviation(magnitudeoftheresultantofdeviationsinxdirections)oftheend-effectorEfromtherigidmotion.Theinitialdesignis50mmfovariablesD.kiInthisexample,theequivalentconstraintsareformedbyemployingthemostcriticalconstraintsandtheresultsarecomparedbyusingtheKreisselmeier-Steinhauserfunction.Inthelatter,differentvaluesofchavebeentried.Ithasbeenobservedthavaluesofcresultedinhighlyconservativedesigns,asexpected.Avalueofc=50yieldedasatisfactorydesign.Itshouldbenotedthatthecompilerlimitsmaybeexceededforlaofcduetotheexponentialfunctionifthelowerboundsondesignv