Non-linear complementary filter on so3: Corrected equations?

Question

While reading the paper "Multirotor Aerial Vehicles: Modeling, Estimation, and Control of Quadrotor" by Mahony, Kumar and Corke, I stumbled across the following equations for a non-linear attitude observer, which I would like to implement, but I believe there is something wrong.

$\dot{\hat{R}} := \hat{R} \left( \Omega_{IMU} - \hat{b} \right)_\times - \alpha \\ \dot{\hat{b}} := k_b \alpha \\ \alpha := \left( \frac{k_a}{g^2}((\hat{R}^T \vec z) \times a_{IMU}) + \frac{k_m}{|^Am|^2} ((\hat{R}^T {^Am}) \times m_{IMU}) \right)_\times + k_E \mathbb{P}_{so(3)} (\hat{R} R_E^T)$

Where $\hat{R}$ and $\hat{b}$ are etimates of orientation and gyroscope bias, $\Omega_{IMU}, a_{IMU}, m_{IMU}, R_E^T$ are measurements and $k_X$ are scalar gains, which may be set to 0 for measurements that are not evailable.

Now $\dot{\hat{R}}$ and $\alpha$ need to be matrices $\in \mathbb{R}^{3\times 3}$ due to their definitions. $\hat{b}$ and thus $\dot{\hat{b}}$ need to be vectors $\in \mathbb{R}^3$. But then what is the correct version of the second equation $\dot{\hat{b}} := k_b \alpha$?

Answer 1

I've implemented this algorithm before but I found a different paper easier to read. Try find the paper by Hamel and Mahoney (with Hamel listed as the first author).

From that paper, which I don't have available at the moment, I believe the first equation should was $$ \begin{align} \dot{\hat{R}} &= \hat{R} \left ( \Omega_{IMU} - \hat{b} - \alpha \right )_\times \end{align} $$ but in that paper $\alpha$ ($\omega$ in the paper) was a vector.

Another thing to look at for the equations you have listed is if the second equation shouldn't have the operator transforming $\alpha$ from its skew-symmetric matrix form to a vector on $\mathbb{R}^3$.