Rotation angles in QTM

The rotation angles of a rigid body can be defined in any number of ways, see chapter Euler angles. The Qualisys standard of the rotation angles are defined as:

  • Rotation around the X-axis is called roll.

  • Rotation around the Y-axis is called pitch.

  • Rotation around the Z-axis is called yaw.

  • Positive rotation is defined as clockwise rotation when looking in the direction of the axis

  • The angles are applied to the local coordinate system in the order: roll, pitch and finally yaw. Therefore to find the rotation of a rigid body with given roll, pitch and yaw angles from QTM, apply the rotations in the same order: first roll, then pitch and finally yaw.

  • QTM uses the following default ranges for the angles, see also the figures below:

    • In these ranges, roll, pitch and yaw are unambiguous and can describe any orientations of a rigid body. 

When the pitch (f) is close to ±90°, small changes in the orientation of the measured rigid body can result in large differences in the rotations because of the singularity at f=±90°, see chapter Rotation angle calculations in QTM.

Below follows an example to show the definitions of the rotation angles. It starts with a 6DOF body, which is in alignment with the global coordinate system.

First the local coordinate system is rotated around the X-axis (roll) with an angle q to the new positions y’ and z’ of the Y- and Z-axis.

After the roll the local coordinate system rotates around the Y-axis (pitch) with the Y-axis in its new position. The X- and Z-axis is rotated with an angle f to the new positions x’ and z’.

Finally the local coordinate system is rotated around the Z-axis (yaw) with the Z-axis in its final position. The X- and Y-axis is rotated with an angle y to the new positions x’ and y’.

After the rotations the rigid body has a new orientation in reference to the global coordinate system, see figure below.

Another description of rotations is to use the rotation matrix, which does not have a singularity. QTM uses the rotation matrix internally to describe the rotation of rigid bodies, and when exporting 6DOF to TSV files the rotation matrix is included for all bodies in all frames, together with roll, pitch and yaw angles. For a description of the calculation of the angles from the rotation matrix, see chapter Rotation angle calculations in QTM.