Euler angles

The sequence is part of the data#

Three angle values are insufficient without an axis order and a statement of whether rotations use fixed axes or axes that move with the body. Reordering the same rotations usually produces a different orientation.

MIT's manipulation notes use roll-pitch-yaw as a specific example: roll about x, pitch about y, and yaw about z, with the extrinsic X-Y-Z convention. “Extrinsic” means rotations about fixed reference axes. “Intrinsic” uses the moving axes.

The SciPy interface, for example, requires the axis sequence and distinguishes intrinsic from extrinsic rotations. The documented sequence is more informative than the generic label.

Gimbal lock is a coordinate singularity#

For the roll-pitch-yaw convention discussed by MIT, at a pitch of 90 degrees, roll and yaw become indistinguishable. Multiple angle combinations describe the same orientation, making the inverse coordinate map singular.

This does not mean the rigid body physically loses a rotational freedom. A robot's kinematic singularity is a separate property of its joint-to-motion mapping.

Choosing another representation#

A rotation matrix or unit quaternion avoids this angle-coordinate singularity. Angle displays can still be convenient for people, provided software treats wraparound, axis order, and singular orientations explicitly.