# Meaning of modified Madgwick Filter equations

I have been implementing the algorithm in the paper titled Design of a Modified Madgwick Filter for Quaternion-Based Orientation Estimation Using AHRS by Amjed S. Al-Fahoum and Momtaz S. Abadir (thank you IJCEE for being open access). I have been able to figure out most parts, but am confused by equations 15 to 17 on page 5 (178 in Journal). They reference Mahoney et al., but that paper is much more opaque to me. I asked the authors recently, but no response yet (if you're reading, you're welcome to answer this here for all).

The equations define $$\omega_{error,t}$$ as a quaternion estimate of gyroscope error, $$\omega_{b,t}$$ as the DC component of $$\omega_{error,t}$$, and $$\omega_{c,t}$$ as the compensated gyroscope measurement:

$$\omega_{error,t} = 2.\hat{q}^*_{est,t-1} \otimes \dot{q}_{est,t} \tag{15}$$

where $$\hat{q}^*_{est,t-1}$$ is the conjugate of the previous time's orientation, and $$\dot{q}_{est,t}$$ is an estimated rate of change of the orientation quaternion from the current and previous estimates.

$$\omega_{b,t} = \sigma\sum{\omega_{error,t}\Delta t} \tag{16}$$

where $$\Delta t$$ is the time since the previous estimate and $$\sigma$$ is a gain parameter.

$$\omega_{c,t} = \omega_{b,t} - \omega_{error,t} \tag{17}$$

1. Why does the error equation hava a different coefficient from the quaternion derivative equation $$\dot{q}=\frac12 \omega q$$?
2. Why aren't the gyroscope measurements ($$\omega_t$$ or $$\dot{q}_{\omega,t}$$) explicitly included in the bias calculation?
• This seems important since $$\omega_{c,t}$$ is used in place of gyroscope measurements $$\omega_t$$.

Equation 17 should read: $$\omega_{c,t} = \omega_{t} - \omega_{b,t} \tag{17}$$
This aligns more with my expectations, but the factor of two in Equation 15 is unexpected to me. It could be absorbed into the $$\sigma$$ in any case, so perhaps it is just a scaling.