Rotation Roll-Pitch-Yaw vs Euler Angle

• Euler angles refer to the angles in a sequence of rotations in a body-fixed frame.
• Roll-Pitch-Yaw angles as a sequence of rotations relative to the space frame.

Example §

• Rotation Matrix of camera frame respect to imu/body frame

$$R_{camera}^{imu}=\left[\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right]$$

• Intrinsic: [ x: -90, y: 90, z: 0 ] === $R_x{R}_y{R}_z$ EulerXYZ about the body frame
• Extrinsic: [ x: -90, y: 0, z: -90 ] === $R_z{R}_y{R}_x$ EulerZYX about the fixed frame
• Both start from axis-X first, then axis-Y then axis-Z

Code §

Eigen::AngleAxisd rotation_x = Eigen::AngleAxisd(-0.5 * M_PI, Eigen::Vector3d::UnitX());
Eigen::AngleAxisd rotation_y = Eigen::AngleAxisd(0.5 * M_PI, Eigen::Vector3d::UnitY());
Eigen::AngleAxisd rotation_z = Eigen::AngleAxisd(-0.5 * M_PI, Eigen::Vector3d::UnitZ());

cout << "[FIXED/EULER] Rotation: " << endl
<< std::setprecision(2) << rotation_z.matrix() * rotation_x.matrix() << endl;

cout << "[roll-pitch-yaw] Rotation: " << endl
<< std::setprecision(2) << rotation_x.matrix() * rotation_y.matrix() << endl;

https://web.mit.edu/2.05/www/Handout/HO2.PDF https://math.stackexchange.com/a/1681118/1133515