User Manual

11.7 Orientation Output Modes

The orientation, calculated by the MTw is the orientation of the sensor-fixed co-ordinate

system (S) with respect to a Cartesian earth-fixed co-ordinate system (G). The output

orientation can be presented in different parameterizations:

 Unit Quaternions (also known as Euler parameters)

 Euler angles

: 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.

11.7.1 Quaternion Orientation Output

A unit quaternion vector can be interpreted to represents a rotation about a unit vector n

through an angle α.

( ( ), ( ))

q cos sin



 n

A unit quaternion itself has unit magnitude, and can be written in the following vector

format;

= (q

)

|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

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 is close to ±90 deg. This singularity is in no way present in the quaternion or rotation

matrix output mode.