Quaternions

Quaternions are hypercomplexnumbers composed of a realand a vector parts

Q q q 9 1 9 It 9e It 9deVectorpart reatorscalarpart

Thequaternion algebrastates that

ii É É 1

II E IIThemultiplicationordermatters

Multiplication table

X I I I II I I I II I I E II I I IitI I I I

Giventhatthemultiplication oftwoquaternionsyield

Q E TUQ Ee qQQ E M EctNrE E t MM t I E NaE14 1 EmIr EeE car It dayItEezII think t I E NaEEn En EnEn EeEu1 EuEu EuCryI ExleyEuEndátlendayEuCadetNhatNEALE

EiEz EFEQQ NNa EIE q Ea NE EFE

Realpart VectorpartHence theresultof quaternion multiplication is identical totheresult of rotationcompositionusingtheEulersymmetric parameters Thus wecanusequaternions torotate referencesystems

tochangebasisNotei Thequaternionmultiplicativeinverse is theconjugatequaternion É 1 E

QÉ MeE 4 E p Etc NE µ EÉ 1 aAssumingthattheµ quaternionnorm is 1Q OI it 191 1 1

Basistransformation using quaternions

let Iaand Isbetwo referencesystem relatedbytheEulersymmetricparameters1411 letI bea vector in whichitsrepresentation in Ia isknown Ya Wewanttofind Is usingquaternions

Considerthefollowingoperation s a Ia QLotta Quaternionwithrealpartoandvectorpartva

a YaQ 1 ELotta q E 4 E loq v.atE t 0 E que VÁE11 E I YEE Na VÁE

vector

VÍ E Et µ até E MIMVatv.ae VÍ E E El year Ya E1Iate YETIa E aíE viva VáE KATEE Ela EVÁEVEIA t 1 a E Ia Ia EIa t E EIa LEYa Ia

Aaté EÉ fvá EE I e EE EEEsyYa 21ÉYa 1 EEtta E EIa ETEIaq ÉE Ya t 2 EEtta 2ME IaIII EE t 2E É 21E Ia

Thus

QYaQ IM EtE 2E É 21E Ia NoteTherealpartofthequaternionÉYaE willalwaysbe 0CbaQYaQ ÉYaQ Csala Ib

Ib ÉYaQRotationcompositionusingquaternions

Letesabethequaternionthattransforms Iainto Is andQasbethe quaternionthattransformsEsmto E Thus thequaternion thattransforms IaintoFecanbecomputedusingYb ÉbalaQue Y y CRsIc EcbIbEcb

Yc ÀcbÉbaIa QbaQdoHowever Ia QcalaÉca Sincethisresultisvalid TI ER then

Eca QsaQas Notice thatthemultiplicationorderis inverseofthatoftheDCMs