Consider a right-hand coordinate system, N with basisvectors $\{\hat{x_N},\hat{y_N},\hat{z_N}\}$ where:
- the x-axis is pointing north
- the y-axis is pointing east
Consider an airplane with a right-hand coordinate system, B with basis vectors $\{\hat{x_B},\hat{y_B},\hat{z_B}\}$ where:
- the x-axis is pointing towards the nose of the plane
- the y-axis is pointing towards the right wing
Looking at an online Euler Angle Visualization Tool it seems that the definition of roll, pitch and yaw of the plane is ?:
- the N basisvectors are first rotated yaw around the the z-axis
- the new basisvectors are rotated pitch around the new y-axis
- the new basisvectors are rotated roll around the new x-axis and the result is the B basis vectors:
$$
\begin{bmatrix} \hat{x_B} \\ \hat{y_B} \\ \hat{z_B} \end{bmatrix}
=R_{BN}*\begin{bmatrix} \hat{x_N} \\ \hat{y_N} \\ \hat{z_N} \end{bmatrix}
$$
where
$$
R_{BN}=RX_{BN}(roll)*RY_{BN}(pitch)*RZ_{BN}(yaw)
$$
I find that
$$
RZ_{BN} = RZ^T
$$
$$
RY_{BN} = RY^T
$$
$$
RX_{BN} = RX^T
$$
$$
R=RZ*RY*RX=R_{BN}^T
$$
where RX, RY RZ and R are the standard rotation matrices used to convert a position from a rotated coordinate system to an unrotated:
Is this correct?
Next let us assume that roll, pitch and yaw is measured by an instrument (an Inertial Measurement Unit). We introduce a coordinate system for the instrument, I with basisvectors $\{\hat{x_I},\hat{y_I},\hat{z_I}\}$.
The instrument is rotated wrt to the airplane: $$ \begin{bmatrix} \hat{x_I} \\ \hat{y_I} \\ \hat{z_I} \end{bmatrix} =R_{IB}*\begin{bmatrix} \hat{x_B} \\ \hat{y_B} \\ \hat{z_B} \end{bmatrix} $$
Given a known $R_{IB}$ and a measurement from the instrument: $(yaw_i, pitch_i,roll_i)$ how can I find $(yaw, pitch,roll)$ of the airplane?