Attitude Rpresentation
Coordinate transformation: map a vector expressed in the coordinate frames $\mathcal{I}$ to the same vector expressed in $\mathcal{B}$, i.e. map $_I\vec{r}$ to $_B\vec{r}$
\[\begin{align} & \begin{pmatrix} \vec{e}^B_x & \vec{e}^B_x & \vec{e}^B_x \end{pmatrix} \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} \\ = & \begin{pmatrix} \vec{e}^I_x & \vec{e}^I_x & \vec{e}^I_x \end{pmatrix} \begin{pmatrix} _I\vec{e}^B_x & _I\vec{e}^B_x & _I\vec{e}^B_x \end{pmatrix}_{3 \times 3} \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} \end{align}\]linear transformation: map a vector to another vector which both are expressed in the same coordinate frame (i.e. $\mathcal{I}$)
\[\begin{align} & \begin{pmatrix} \vec{e}^I_x & \vec{e}^I_x & \vec{e}^I_x \end{pmatrix} \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} \\ \rightarrow & \begin{pmatrix} \vec{e}^I_x & \vec{e}^I_x & \vec{e}^I_x \end{pmatrix} \begin{pmatrix} _I\vec{e}^B_x & _I\vec{e}^B_x & _I\vec{e}^B_x \end{pmatrix}_{3 \times 3} \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} \end{align}\]If $q$ or $R$ represent coordinate transformation from IMU to WORLD,
- then their correponding rodregues is meaning that find a axis and angle to rotate IMU frame align to WORLD frame
- their corresponding Euler angle (xyz-rpy) is meaning that rotate IMU frame’s x axis (roll) and so on, finally aligning to WORLD frame.