The above equation is used to perform rotations using unit quaternions. All along, the requirement of preserving vector magnitude never explicitly forced us to choose unit quaternions. However, this has been ensured because we started with a vector equation that took this into account and unit quaternion was an output of subsequent manipulations of this equation. <br \>

Formulation of quaternions has made two major accomplishments. Firstly, it was able to give Euler’s rotation theorem (which says any rotation of a ‘rigid body’ is equivalent to a single rotation about a unique axis) a more solid mathematical application. Secondly, it was able to “mathematically” resolve the problem of gimbal lock ([https://www.youtube.com/watch?v=zc8b2Jo7mno here's a nice video on gimbal lock]) that euler angles face. It should be pointed out now that though quaternions provided a ‘mathematical’ alternative to euler angles, they didn’t fully dislodge them. For instance, there are still mechanical systems that perform successive rotations in a predefined manner to realise a final rotation. It is natural and convenient to describe these using euler rotation angles. Quaternions in this case can’t help circumvent the gimbal lock issue because the euler angles are a physical requirement. Quaternions are, hence, only a ‘mathematical’ antidote to gimbal lock. Nevertheless, they are a huge simplification of their existing mathematical counterparts. <br \>

If you are done reading this page, you can go back to [[Attitude Determination and Control Subsystem]]