# Which way of integrating the quaternion representation of orientation is correct?

I am wondering how to integrate a measurement of angular velocity into an orientation quaternion. Suppose I have measurements of the rotation rate about the three principal axes of a body. I would combine them like so:

$${^B_R}{\dot{q}}{_{\omega,t}}=\frac{1}{2}{^B_R}{q}{_t}\otimes[0,\omega{_x},\omega{_y},\omega{_z}] ,$$

where $$\omega{_x},\omega{_y},\omega{_z}$$ are the angular rates about the $$x$$-,$$y$$- and $$z$$-axis (e.g. in deg/sec). Also, the leading sub-script $$R$$ denotes the inertial reference frame while $$B$$ denotes the body frame. A leading sub-script denotes the frame being described and a leading super-script denotes the frame this is with reference to.

Now I have seen (and I thought so myself) that people use

$${^B_R}{q}{_{t+1}}={^B_R}{q}{_{t}}\otimes({^B_R}{\dot{q}}{_{\omega,t}}\Delta{t})$$

to compose the two rotations. However, the paper I'm following and that I want to implement just adds them up:

$${^B_R}{q}{_{t+1}}={^B_R}{q}{_{t}}+{^B_R}{\dot{q}}{_{\omega,t}}\Delta{t}$$

I am aware that this might not be the right site, I thought about maths or scicomp, but I was unsure. Also I can see that the linked question seems like an answer, but I am more looking for an explanation for why the author (Sebastian Madgwick) used the additive operation instead of composition.