In many application, the knowledge of the orientation of a body is essential for control purposes. There are several representations, where one is the Euler Angles and another is the Quaternion represenation, where the Euler Angles suffer from gimbal lock. My question is, how can one use the Quaternion represenation for control the angular position in space? If this would be possible, then why are the Euler Angles so widespread?
Euler angles are easier to understand and use. Imagine a airtrafic controller getting a aircraft heading info as quaternion data. Euler angles are significantly easier to unerstan interpret and interpolate.
While quaternions do have benefits they are also conceptually more complex to work with. In many applications the downsides of euler angles and matrices are of no concern.
In many mechanical cases you can not avoid gimbal lock anyway, having a control system that can navigate a problem area your physical counterpart can not is not very useful. All you end up with is diverging from your real system such as a industrial robot.
Similar thig applies to simulation where the euler-larange equation just sidesteps the issue. The euler angle representation can represent all orientations just not interpolate well in this nonuniform space, but since the larangian is not operating in this space it matters not and the solution is well known. No gimbal lock concerns. Altough quats have a very small advantage in speed of normalisation. But then all formulas are usually in angular or matrix form.
Third, while accumulating number of revolutions is usually futile, its viable to do in euler angles for the kinds of angles that dont come into question. While gimbal lock is nasty, it comes up quite rarely.
I think joojaa summed it up well, Euler angles are fairly intuitive, given a set of angles you can quickly get an idea of what the object's orientation is. Gimbal lock does present a problem especially in video games and some applications in biomechanics/robotics. Quaternions are frequently used when gimbal lock is an issue.
Luckily one can convert between quaterions and Euler angles, so you could have a control system that internally runs on quaternions but reports data as Euler angles if the user needs to look at what is happening.