Derivative of quaternion
When I was searching for uses of quaternions in orientation algorithms, there was always one equation that was shown with no explanation: \begin{gather}\dot{\mathbf{q}}(t)=\frac{1}{2}\boldsymbol{\omega}\mathbf{q}(t) \end{gather} but sometimes it was: \begin{gather}\dot{\mathbf{q}}(t)=-\frac{1}{2}\boldsymbol{\omega}\mathbf{q}(t) \end{gather} or \begin{gather}\dot{\mathbf{q}}(t)=-\frac{1}{2}\mathbf{q}(t) \boldsymbol{\omega}\end{gather} Moreover, when I was looking for any derivation of these formulas I found out that all resources were several pages - not very encouraging for me or derivations were not complete in their own calculations, which was even more frustrating when you were thinking that finally, you would understand this. So, I decided to do it myself in 2 ways: first more based on quaternion interpretation and the second old-fashion derivative from the definition. Also, I will discuss frames of reference in which all of the elements are presented. I will provide some transfo