Rotation Sequence Is of Major Importance in Establishing the Equivalence Between Quaternion

Arland Thompson

Chief Scientist

Advanced Technology Associates

(www.atacolorado.com)

Rotation sequence is of major importance in establishing the equivalence between quaternion elements and Euler angles. A second equally important consideration is the quaternion convention adopted.

The notation used in this document is that all vectors will be denoted by an underscore, all matrices will be denoted by a double underscore, and quaternions are denoted by underscore and capitalization. The quaternion convention followed here is scalar part last. The quaternion components are one-based. The notation for Euler angles used here is conventional in the sense that is the first angle in the sequence, is the second angle in the sequence, and is the third angle in the sequence.

=T. Euler angle sequence

For example, consider first the Elementary Rotations (123 sequence)

= = =

The corresponding quaternions are:

Roll, , About Pitch, , About Y Yaw, , About Z

Euler sequences divide naturally into two classes; type I sequences have no repeating axes (e.g. 123); type II sequences repeat the external axes (e.g. 131). There are 12 possible Euler sequences (123, 132, 213, 231, 312, 321, 121, 131, 212, 232, 313, and 323). There exists a two-fold ambiguity in the internal rotation angle for both sequences. Because of the ambiguity in the internal angle, there are 24 possible Euler sequences. For a type I sequence, the ambiguity is typically resolved by choosing to be between -90 and 90 degrees, which gives the cosine of the angle to be >= 0. For a type II sequence, the ambiguity is usually resolved by choosing sin to be >= 0, which puts between 0 and 180 degrees. Singularities also exist for both types of sequences. For type I sequences, singular values occur at multiples of 180 degrees for the internal angle. For a type II sequence, singular values occur at odd multiples of 90 degrees for the internal angle.

The following gives the corresponding direction cosine matrix (DCM) for each of the 12 possible Euler sequences. The direction of the transformation represented by the DCM is from the stationary frame to the rotated frame. In other words, a vector in the stationary frame can be expressed in the rotated frame by pre-multiplying the vector by the DCM. The quaternion corresponding to the Euler sequence is also given, along with the Euler angles derived from the DCM. This is repeated for the transpose of the DCM. In both cases (DCM and transpose of the DCM), the Euler angles are computed as functions of the quaternion components.

Note: When Euler angles are computed from the inverse DCM, these are the original Euler angles to go from the stationary frame to the rotated frame. To get the Euler angles to go from the rotated frame to the stationary frame, just negate the angles and reverse the order.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Type I sequences

1. Roll – Pitch – Yaw Sequence , (X,Y,Z)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

=

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

- - - - - - - - - - - - - - - - - - - - - - - - - -

2. Roll – Yaw - Pitch Sequence , (X, Z, Y)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

- - -

- - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

- - -

3. Pitch – Roll – Yaw Sequence (Y, X, Z)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

4. Pitch – Yaw –Roll Sequence (Y, Z, X)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

5. Yaw – Roll – Pitch Sequence (Z, X, Y)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

6. Yaw – Pitch – Roll Sequence , (Z, Y, X)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

- - - - - - - - - - - - - - - - - - - - - - - -

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

Type II sequences

7. Roll – Pitch – Roll Sequence (X, Y, X)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - -- - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

8. Roll – Yaw – Roll Sequence (X,Z,X)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

9. Pitch – Roll – Pitch (Y, X, Y)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from Quaternion Inverse

=

=

10. Pitch – Yaw – Pitch (Y, Z, Y)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from DCM and Quaternion inverse

=

=

11. Yaw – Roll – Yaw Sequence (Z, X, Z)

Direction Cosine Matrix from Euler angles (Stationary Frame to Rotated Frame)

Quaternion from Euler Angles (Stationary Frame to Rotated Frame)

DCM from Quaternion (Stationary Frame to Rotated Frame)

Euler Angles from DCM and Quaternion

=

=

- - - - - - - - - - - - - - - - - - - - -

Inverse of Quaternion (Rotated Frame to Stationary Frame)

DCM from Quaternion (Rotated Frame to Stationary Frame)

Euler Angles from DCM and Quaternion inverse

=

=

12. Yaw – Pitch – Yaw Sequence (Z,Y,Z)

DCM from Quaternion (Stationary Frame to Rotated Frame)

=

=

- - - - - - - - - - - - - - - - - - -

DCM from Quaternion (Rotated Frame to Stationary Frame)

=

=

References:

Spacecraft Attitude Determination and Control; Edited by James R. Wertz