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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 $\omega_t$.
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Reply from one of the authors:

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.

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