User manual


 DocumentMT0100P.N
©XsensTechnologiesB.V. MTiandMTxUserManualandTechnicalDocumentation
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15
4.3.3 RotationMatrixorientationoutputmode
Therotationmatrix (alsoknown asDirection CosineMatrix, DCM)isawell‐known,redundantand complete
representation of orientation. The rotation matrix can be interpreted as the unit‐vector components of the
sensorcoordinatesystemSexpressedinG.ForR
GS
theunitvectorsofSarefoundinthecolumnsofthematrix,
socol1isX
S
expressedinGetc.Arotationmatrixnormisalwaysequaltoone(1)andarotationR
GS
followed
bytheinverserotationR
SG
naturallyyieldstheidentitymatrixI
3
.

1R =
RR

GS SG
=
3
I
2222
22 22q q q q qq qq qq qq
⎡⎤
+−− − +
+
ZY X
RRRR


Therotationmatrix,R
GS
,canbeinterpretedintermsofquaternions;

01 23 12 03 02 13
2222
03 12 0 1 2 3 23 01
2222
13 02 23 01 0 1 2 3
22
01 12031302
22
12 03 0 2 23 01
13
22 22
22 2 2
2212 2 2 2
22 22122
22
GS
Rqqqqqqqqqqqq
qq qq qq qq qqqq
q q qq q q qq q q
qq q q q q q q qq
qq q
⎢⎥
=+ −+− −
⎢⎥
⎢⎥
−+−−
⎣⎦
+− − +
=+ +− −
−
22
02 23 01 0 3
22221qqqqqqq
⎡⎤
⎢⎥
⎢⎥
⎢⎥
++−
⎣⎦


orintermsofEuler‐angles;

cos sin 0 cos 0 sin 1 0 0
sin cos 0 0 1 0 0 cos sin
001sin0cos0sincos
cos cos sin sin cos cos sin cos sin cos sin sin
cos sin sin s in sin cos cos cos sin si
GS
ψθφ
ψψ θ θ
ψψ φ φ
θθφφ
θ
ψφθψφψ φθψφψ
θψ φθψ φψ φθ
−
⎡⎤⎡⎤⎡⎤
⎢⎥⎢⎥⎢⎥
=−
⎢⎥⎢⎥⎢⎥
⎢⎥⎢⎥⎢⎥
−
⎣⎦⎣⎦⎣⎦
−+
=+nsincos
sin sin cos cos cos
=
ψ
φψ
θφθ φθ
⎡⎤
⎢⎥
−
⎢⎥
⎢⎥
−
⎣⎦



Asdefinedhere

R
GS
,rotatesavectorinthesensorco‐ordinatesystem(S)totheglobalreferencesystem(G):

()
T
RR==
S
xx x
ad g R R R
⎡⎤⎡ ⎤
⎥
⎥
⎥
⎦
GS SGGS


Itfollowsnaturallythat,R
SG
rotatesavectorintheglobalreferenceco‐ordinatesystem(G)tothesensorco‐
ordinatesystem(S).

Fortherotationmatrix(DCM)outputmodeitisdefinedthat:

11 12 13
21 22 23
31 32 33
GS
RbehRRR
cf i R R R
⎢⎥⎢
==
⎢⎥⎢
⎢⎥⎢
⎣⎦⎣
