Document MT0605P.E
© Xsens Technologies B.V.
MTi User Manual
32
Euler angles
7
: roll, pitch, yaw (XYZ Earth fixed type, also known as Cardan or aerospace
sequence)
Rotation Matrix (directional cosine matrix)
A positive rotation is always “right-handed”, i.e. defined according to the right hand rule (corkscrew
rule). This means a positive rotation is defined as clockwise in the direction of the axis of rotation.
NOTE:
This section is intended to give detailed information on the definition of the various orientation
output modes of the MTi. The output sequence of the elements in the vectors and matrices defined
here holds for all interface options (Low-level communication protocol, API, GUI). For more detailed
information about the respective interfaces please refer to their specific documentation;
Low-level communication
MTi Low-level Communication Documentation
GUI
MT Manager User Manual
4.5.1
Quaternion orientation output mode
A unit quaternion vector can be interpreted to represents a rotation about a unit vector
n
through an
angle
α
.
𝑞
𝐿𝑆
= (cos (
𝛼
2
) , 𝒏 sin (
𝛼
2
))
A unit quaternion itself has unit magnitude, and can be written in the following vector format;
𝑞
𝐿𝑆
= (𝑞
0
, 𝑞
1
, 𝑞
2
, 𝑞
3
)
|𝑞 | = 1
Quaternions are an efficient, non-singular description of 3D orientation and a quaternion is unique up
to sign:
𝑞 = −𝑞
An alternative representation of a quaternion is as a vector with a complex part, the real component is
the first one, q
0
.
The inverse (
q
SL
) is defined by the complex conjugate (
†
) of
q
LS
. The complex conjugate is easily
calculated;
𝑞
𝐿𝑆
†
= (𝑞
0
, −𝑞
1
, −𝑞
2
, −𝑞
3
) = 𝑞
𝑆𝐿
7
Please note that 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 earth-fixed reference frame (i.e. pitch approaches ±90
). In practice this means
roll and pitch is not defined as such when pitch is close to ±90 deg. This singularity is in
no way
present in the
quaternion or rotation matrix output mode.
