EoMs · Euler transforms

Aircraft attitude: the 3-2-1 Euler sequence

An aircraft's orientation is described by three ordered rotations from an Earth axis system (North-East-Down) to the body axes: yaw ψ about Earth down, then pitchθ, then roll φ. Move the sliders to set an attitude and watch the body triad and aircraft follow. The direction-cosine matrix and orientation quaternion below are recomputed in Rust compiled to WebAssembly on every change.

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Yaw ψ0°

Pitch θ0°

Roll φ0°

x_b — forward y_b — starboard z_b — down

Earth → Body DCM & quaternion (from WASM)

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q = [x, y, z, w]

What you're looking at

The grey arrows are the fixed Earth axes (N, E, D). The coloured triad is the body frame:x_b forward,y_b out the starboard wing,z_b down through the belly.
The matrix is the Earth→Body DCM: C·v_earth = v_body. Its rows are the body axes expressed in Earth coordinates — exactly what positions the triad.
Push pitch to ±90° and the yaw and roll axes line up: that's gimbal lock, the singularity the quaternion avoids.