User Manual

© Xsens Technologies B.V.
66
The inverse (q
SG
) is defined by the complex conjugate (†) of q
GS
. The complex conjugate can
be calculated:
†
0 1 2 3
( , , , )
GS SG
q q q q q q    
As defined here q
GS
rotates a vector in the sensor co-ordinate system (S) to the global
reference co-ordinate system (G).
†
GS GS GS SG
q q q q
G S S
x x x
Hence, q
SG
rotates a vector in the global reference co-ordinate system (G) to the sensor co-
ordinate system (S), where q
SG
is the complex conjugate of q
GS
.
11.7.2 Euler Angles Orientation Output Mode
The definition used for 'Euler-angles' here is equivalent to 'roll, pitch, yaw/heading' (also
known as Cardan). The Euler-angles as orientation output are provided as XYZ Earth fixed
type (subsequent rotation around global X, Y and Z axis, also known as aerospace sequence).
 φ = roll
1
= rotation around X
G
, defined from [-180
o
…180
o
]
 θ = pitch
2
= rotation around Y
G
, defined from [-90
o
…90
o
]
 ψ = yaw
3
= rotation around Z
G
, defined from [-180
o
…180
o
]
NOTE: Due to the definition of Euler angles there is a mathematical singularity (gimbal lock)
when the sensor-fixed X-axis is pointing up or down in the earth-fixed reference frame (i.e.
pitch approaches ±90
o
). This singularity is in no way present in the quaternion or rotation
matrix output mode. The singularity cannot be compensated for but only avoided using the
rotation matrix output, then manually extract Euler Angles by using different Euler
sequences
4
.
The Euler-angles can be interpreted in terms of the components of the rotation matrix, R
GS
,
or in terms of the unit quaternion, q
GS
;
11
32 2 3 0 1
22
33 0 3
11
31 1 3 0 2
11
1 2 0 3
21
22
11 0 1
22
tan tan
2 2 1
sin ( ) sin (2 2 )
22
tan tan
2 2 1
GS
GS
GS
R q q q q
R q q
R q q q q
q q q q
R
R q q






   


   

   
    









1
“roll” is also known as: “bank”
2
“pitch” is also known as: “elevation” or “tilt”
3
“yaw” is also known as: “heading”, “pan” or “azimuth”
4
Woltring HJ. 3-D attitude representation of human joints: A standardization proposal.
Journal of Biomechanics. 1994;27 (12):1399-1414.