rotation sequence is of major importance in establishing the

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1、ArlandThompsonChiefScientistAdvancedTechnologyAssociates(www.atacolorado.com)RotationsequenceisofmajorimportanceinestablishingtheequivalencebetweenquaternionelementsandEulerangles.Asecondequallyimportantconsiderationisthequaternionconventionadopted.Thenotationusedinthisdocumentisthat

2、allvectorswillbedenotedbyanunderscore,allmatriceswillbedenotedbyadoubleunderscore,andquaternionsaredenotedbyunderscoreandcapitalization.Thequaternionconventionfollowedhereisscalarpartlast.Thequaternioncomponentsareone-based.ThenotationforEuleranglesusedhereisconventionalinthesensetha

3、tisthefirstangleinthesequence,isthesecondangleinthesequence,andisthethirdangleinthesequence.=T.EuleranglesequenceForexample,considerfirsttheElementaryRotations(123sequence)===Thecorrespondingquaternionsare:Roll,,AboutPitch,,AboutYYaw,,AboutZEulersequencesdividenaturallyintotwoclasses

4、;typeIsequenceshavenorepeatingaxes(e.g.123);typeIIsequencesrepeattheexternalaxes(e.g.131).Thereare12possibleEulersequences(123,132,213,231,312,321,121,131,212,232,313,and323).Thereexistsatwo-foldambiguityintheinternalrotationangleforbothsequences.Becauseoftheambiguityintheinternalang

5、le,thereare24possibleEulersequences.ForatypeIsequence,theambiguityistypicallyresolvedbychoosingtobebetween-90and90degrees,whichgivesthecosineoftheangletobe>=0.ForatypeIIsequence,theambiguityisusuallyresolvedbychoosingsintobe>=0,whichputsbetween0and180degrees.Singularitiesalsoexistfor

6、bothtypesofsequences.FortypeIsequences,singularvaluesoccuratmultiplesof180degreesfortheinternalangle.ForatypeIIsequence,singularvaluesoccuratoddmultiplesof90degreesfortheinternalangle.Thefollowinggivesthecorrespondingdirectioncosinematrix(DCM)foreachofthe12possibleEulersequences.Thed

7、irectionofthetransformationrepresentedbytheDCMisfromthestationaryframetotherotatedframe.Inotherwords,avectorinthestationaryframecanbeexpressedintherotatedframebypre-multiplyingthevectorbytheDCM.ThequaternioncorrespondingtotheEulersequenceisalsogiven,alongwiththeEuleranglesderivedfrom

8、theDCM.Thisi