Representing-Attitude-Euler-Angles-Unit-Quaternion

RepresentingAttitude:EulerAngles,UnitQuaternions,andRotationVectorsJamesDiebelStanfordUniversityStanford,California94301{9010Email:[e-mail]20October2006AbstractWepresentthethreemainmathematicalconstructsusedtorepresenttheattitudeofarigidbodyinthree-dimensionalspace.Theseare(1)therotationmatrix,(2)atripleofEulerangles,and(3)theunitquaternion.Totheseweaddafourth,therotationvector,whichhasmanyofthebene¯tsofbothEuleranglesandquaternions,butneitherthesingularitiesoftheformer,northequadraticconstraintofthelatter.Thereareseveralothersubsidiaryrepresentations,suchasCayley-Kleinparametersandtheaxis-anglerepresentation,whoserelationstothethreemainrepresentationsarealsodescribed.Ourexpositioniscateredtothosewhoseekathoroughanduni¯edreferenceonthewholesubject;detailedderivationsofsomeresultsarenotpresented.Keywords{Eulerangles,quaternion,Euler-Rodriguesparameters,Cayley-Kleinparameters,rotationmatrix,di-rectioncosinematrix,transformationmatrix,Cardanangles,Tait-Bryanangles,nauticalangles,rotationvector,orientation,attitude,roll,pitch,yaw,bank,heading,spin,nutation,precession,Slerp1Contents1Introduction41.1OverviewofContents..............................................41.2Sources......................................................41.3CoordinateSystems...............................................42RotationMatrix42.1CoordinateTransformations...........................................52.2TransformationMatrix..............................................52.3PoseofaRigidBody...............................................52.4CoordinateRotations..............................................52.5DirectionCosineMatrix.............................................52.6BasisVectors...................................................62.7RotationMatrixMultiplication.........................................63Kinematics63.1Notation......................................................63.2MotionofaFixedPointonaRigidBody...................................73.3MotionofaParticleinaMovingFrame....................................74FiniteDi®erenceApproximations75EulerAngles75.1RotationSequence................................................75.2EulerAngleRatesandAngularVelocity....................................95.3Linearization...................................................95.4ValidRotationSequences............................................95.5EulerAngleSequence(3,1,3)..........................................105.5.1Usage...................................................105.5.2EulerAngles)RotationMatrix....................................105.5.3EulerAngles(RotationMatrix....................................105.5.4EulerAngles)EulerAngleRatesMatrices..............................105.5.5EulerAngles)UnitQuaternion....................................115.5.6Singularities................................................115.6EulerAngleSequence(1,2,3)..........................................115.6.1Usage...................................................115.6.2EulerAngles)RotationMatrix....................................115.6.3EulerAngles(RotationMatrix....................................125.6.4EulerAngles)EulerAngleRatesMatrices..............................125.6.5EulerAngles)UnitQuaternion....................................125.6.6Singularities................................................135.7DerivativesofSelectedTrigonometricFunctions................................135.8Singularities....................................................135.9Intra-Euler-AngleConversion..........................................135.9.1Sequence(3,1,3)(Sequence(1,2,3)..................................135.9.2Sequence(1,2,3)(Sequence(3,1,3)..................................136Quaternions146.1GeneralQuaternions...............................................146.2QuaternionMultiplication............................................146.3Quaternion)QuaternionMatrices......................................146.4UnitQuaternion)RotationMatrix......................................156.5UnitQuaternion(RotationMatrix......................................156.6QuaternionRates)AngularVelocity.....................................166.7QuaternionRates(AngularVelocity.....................................166.8QuaternionRates)AngularAcceleration...................................166.9QuaternionRates(AngularAcceleration...................................166.10UnitQuaternion(Cayley-KleinParameters.................................1626.11UnitQuaternion)Cayley-KleinParameters.................................176.12UnitQuaternion(Axis-Angle.........................................176.13UnitQuaternion)Axis-Angle.........................................186.14UnitQuaternion(EulerAngles........................................186.15UnitQuaternion)EulerAngles........................................186.16OptimizationwithQuaternions.........................................187RotationVectorRepresentation187.1RotationVector(Axis-Angle.........................................187.2RotationVecto