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ComputersandElectronicsinAgriculture44(2004)63–69Dynamicweighingofdairycows:usingalumped-parametermodelofcowwalkDraganCveticanina,∗,GeorgWendlbaRemming,HerojaTepica15,21000NoviSad,SerbiaandMontenegrobResearchCentreofAgriculturalEngineering,BavarianStateResearchCentreforAgriculture,VöttingerStr.36,85354Freising,GermanyReceived13June2003;receivedinrevisedform27October2003;accepted1March2004AbstractThepaperdealswiththedynamicweighingofdairycows.Asimplephysical–mathematicalmodelthatsimulatescowwalkisdescribed.Thefour-leggedsystemofcowmovementissimplifiedasatwoleggedsystem.Theoutputofthemodel,whichistheforcetothegroundproducedbyacow’sfeet,dependsonwalkingspeedandbodyweight.Thedevelopedmodelwasusedforbodyweightdeterminationascowscrossedawalk-throughweighingscale.Anaccuracyof2%comparedtostaticallymeasuredbodyweightcouldbeachievedintheabsenceofcrowdingofanimalstraversingtheplatform.©2004ElsevierB.V.Allrightsreserved.Keywords:Dynamicweighing;Cowwalk;Modelling1.IntroductionDairyfarmingisdevelopingintoautomatedproductionsystem.Theobjectiveistofullyautomateeveryprocessfromfeedingtomilkinginordertoreduceproductioncostsandmaximisemilkyield.Dynamicweighingofcattleisanimportantsegmentofthissystem.Weighinginmotionwithoutstoppingtheanimal’smovementafteritleavesthemilkingparlourisofinterestduetocontinuityofthemilkingprocessandsmallervariationsofbodyweightwhenmeasuredatthesametimeoftheday.Theforcestothegroundduringthecow’spassagearerecordedonwalk-throughweighers.Walk-throughweighingscale∗Correspondingauthor.E-mailaddress:cdragan@eunet.yu(D.Cveticanin).0168-1699/$–seefrontmatter©2004ElsevierB.V.Allrightsreserved.doi:10.1016/j.compag.2004.03.00164D.Cveticanin,G.Wendl/ComputersandElectronicsinAgriculture44(2004)63–69designandbodyweightmeasurementmethodshavebeendescribedbyFilbyetal.(1979),Renetal.(1992),andPeiperetal.(1993).Theywerebasedonfilteringthemeasuredforcesignalandaveragingitorrecordingthepeakofthesignalastherelevantvalue.However,forfastcrossingoftheweighingplatform,aswellasforcrowdingofanimalsontheplatform,thesemethodsproducedresultsthatcouldnotbeusedasvalid.Intheworkpresentedhere,forcestothegroundproducedduringcowmovementareanalysedthroughamathematicalmodelthatcansimulatecowlocomotion.Thegoalistofindthevalueofthesummedforcesproducedbyfeetforanywalkingvelocityandbodymass.Themodelisusedtodeterminethebodyweightofthecowcrossingawalk-throughweigher.Theobjectiveistoincreaseaccuracyofdynamicweighinginthecaseoffastcrossingtheplatform.2.ModellingofcowmovementAlumped-parametermodeliscreatedsimilartothesimplehumanmovementmodeldescribedinAlexander(1995),Garciaetal.(1998)andTodd(1985).Theassumptionisthatthefour-leggedanimalsimultaneouslymovesitstwodiagonallegssothatitcanbetreatedasatwoleggedbeing.Thewalkingisassumedtobetherotationoftheactivelegaroundthefootandtherotationofthepassivelegaroundthehip.Thephysicalmodelofsuchsystemisaninvertedpendulumconsistedofalinkbarandapointmassatitsend,afreebarconnectedtothepointmassandamotorthatdrivestherigidbars(Fig.1a).Theanimal’smotionisthussimplifiedtorotationaroundtwojoins,oneatgroundlevelandtheotheratthepointmass.ThepointmassMisthebodymassoftheanimal,whilethetwobarshavethemassmandlengthlofthelegs.ThetorquesT1andT2whichmovethebarsrepresentthemuscularactionofthelegs.Theanglepositionsofthebarsareθandϕ.Thecontrollertracksthedesiredinputangleofrotationofthebars.AsshowninFig.1b,afullstepisdonewhenalinkagelegmovesfrom−θto+θ.Asinwalkingthelegsswitchateverystepandtheactivelegbecomespassive,theanglesθandϕchangetheirnamessuchthatθbecomesϕ,andviceversa.ThesystemhastwodegreesoffreedomandtheLagrangedifferentialequationsofmotionareddt∂EK∂˙θ−∂EK∂θ+∂EP∂θ=T1,(1)ddt∂EK∂˙ϕ−∂EK∂ϕ+∂EP∂ϕ=T2,(2)whereT1andT2aretorqueswhichactonthebars,EKisthekineticenergyEK=(23ml2+12Ml2)˙θ2+16ml2˙ϕ2+12ml2cos(θ+ϕ)˙θ˙ϕ(3)andEPisthepotentialenergyofthesystemEP=mg12lcosθ+Mglcosθ+mg(lcosθ−12lcosϕ).(4)D.Cveticanin,G.Wendl/ComputersandElectronicsinAgriculture44(2004)63–6965T2T1m,lm,lMθϕDesiredanglesθd,ϕdControllingtorqueT1,T2InvertedpendulumwithonefreebarsystemdynamicsController(a)ϕθθϕMMM(1)(2)(3)θϕMMϕθM(4)(5)(6)(b)Fig.1.Thephysicalmodelofcowlocomotion:(a)theinvertedpendulummodelwiththecontrolsystem;(b)twofullstepsofthemodel.Substituting(3)and(4)into(1)and(2)thedifferentialequationsofmodelofcowwalkingare(43m+M)l2¨θ+12ml2cos(θ+ϕ)¨ϕ−12ml2˙ϕ2sin(θ+ϕ)−gl(32m+M)sinθ=T1,(5)12ml2cos(θ+ϕ)¨θ+13ml2¨ϕ−12ml2˙θ2sin(θ+ϕ)+mg12lsinθ=T2.(6)Decouplingthemodel,theforcesactingonthesystemduringwalkingsimulationofthemodelarepresentedinFig.2andtheverticalFNandhorizontalFTforcestothegroundareobtainedFT=m(¨xC1+¨xC2),(7)66D.Cveticanin,G.Wendl/ComputersandElectronicsinAgriculture44(2004)63–69C1C2yxFTFNMgmgmgYMXMYMXMFig.2.Activeforcesandjointsreactionsinthesystem.FN=m(¨yC1+¨yC2)+g(M+2m),(8)wherexC1=(l/2)sinθ,xC2=lsinθ+(l/2)sinϕ,yC1=(l/2)cosθ,yC2=lcosθ−(l/2)cosϕ.Fortheinitialconditionsitisassumedthattheangularpositionofthebarscorrespondstotheendpositionofthepreviousstep.Asthekineticenergyislostininelasticimpactofafootwiththegrounditissupposedthatthevelocitiesofbarsfortheendp