Document MT0605P.E
© Xsens Technologies B.V.
MTi User Manual
33
As defined here
q
LS
rotates a vector in the sensor co-ordinate system (
S
) to the global reference co-
ordinate system (
L
).
𝒙
𝐿
= 𝑞
𝐿𝑆
𝒙
𝑆
𝑞
𝐿𝑆
†
= 𝑞
𝐿𝑆
𝒙
𝑠
𝑞
𝑆𝐿
Hence,
q
SL
rotates a vector in the global reference co-ordinate system (L) to the sensor co-ordinate
system (
S
), where
q
SL
is the complex conjugate of
q
SL
.
4.5.2
Euler angles orientation output mode
Euler angles describe the rotation of a rigid body by means of three successive rotations in a particular
sequence. The Euler angles used are ‘roll, pitch, yaw’, referred to in the literature as Cardan/Tait-
Bryan angles. The sequence of rotations for Euler angles follows the aerospace convention (Z-
Y’-X’’
sequence) for rotation from the global reference co-ordinate system (
L
) to the sensor co-ordinate
system (
S
).
ψ = yaw
8
= rotation around Z
L
, defined from [-180
…180
]
θ = pitch
9
= rotation around Y
L
’ which is the current Y axis after the first rotation, defined from
[-90
…90
]
φ = roll
10
= rotation around X
L
’’, which is the current X axis after the second rotation, defined
from [-180
…180
]
NOTE:
Due to the definition of Euler angles there is a mathematical singularity when the sensor-fixed
x-axis is pointing up or down in the
L
co-ordinate system (i.e. pitch approaches ±90
). This singularity
is not present in the quaternion or direction cosine matrix (rotation matrix) representation. Quaternion
and rotation matrix output modes can be used to access these orientation representations
respectively.
The Euler-angles can be interpreted in terms of the components of the rotation matrix,
R
LS
, or in terms
of the unit quaternion,
q
LS
;
𝜑
𝐿𝑆
= 𝑡𝑎𝑛
−1
(
𝑅
32
𝑅
33
) = 𝑡𝑎𝑛
−1
(
2𝑞
2
𝑞
3
+ 2𝑞
0
𝑞
1
2𝑞
0
2
𝑞
3
2
− 1
)
𝜃
𝐿𝑆
= −𝑠𝑖𝑛
−1
(𝑅
31
) = −𝑠𝑖𝑛
−1
(2𝑞
1
𝑞
3
− 2𝑞
0
𝑞
2
)
ѱ
𝐿𝑆
= 𝑡𝑎𝑛
−1
(
𝑅
21
𝑅
11
) = 𝑡𝑎𝑛
−1
(
2𝑞
1
𝑞
2
+ 2𝑞
0
𝑞
3
2𝑞
0
2
𝑞
1
2
− 1
)
Here, the arctangent (tan
-1
) is the four quadrant inverse tangent function.
NOTE:
that the output is in
degrees
and not radians.
4.5.3
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
L-coordinate system
. For
R
LS
the unit
vectors of
S
are found in the columns of the matrix, so col 1 is
X
S
expressed in
L
etc. A rotation matrix
8
“yaw” is also known as: “heading”, “pan” or “azimuth”
9
“pitch” is also known as: “elevation” or “tilt”
10
“roll” is also known as: “bank”
