Euler Angles and Gimbal Lock

Roll, pitch, and yaw are easy to read, which is why drones and cars report them. But at one special pitch angle, two of the three rotation axes line up, and the three numbers can only describe two kinds of motion. That is gimbal lock.

Yaw ψ (outer ring) Pitch θ (middle ring) Roll φ (inner ring)

Drag the 3D view to look from another angle.

Yaw, pitch, roll

The common aerospace convention applies yaw $\psi$ about z, then pitch $\theta$ about the new y, then roll $\phi$ about the newest x:

$$R = R_z(\psi)\,R_y(\theta)\,R_x(\phi)$$

At $\theta = 90^\circ$ the roll axis points straight along the yaw axis. Then only the difference between yaw and roll changes the pose:

$$R\big|_{\theta = 90^\circ} \text{ depends only on } (\psi - \phi)$$

What to notice

Press "Go to gimbal lock". The red roll axis and the amber yaw axis now point the same way, so turning the yaw slider and the roll slider spin the body about the same axis. One way of turning is simply gone, and nearby the angles change wildly for small motions. This really happened on the Apollo 11 guidance gimbals, which had a warning light for it.

Robots avoid this by storing orientation as a rotation matrix or a quaternion, and only turning it into roll, pitch, and yaw for display.