Euler Angles

Definition

Euler angles encode a rotation as three successive rotations about principal axes, e.g.

$$R(\alpha ,\beta ,\gamma )=R_x(\alpha )R_y(\beta )R_z(\gamma )$$

Only 3 parameters, intuitive, easy derivatives.

Two things must be specified

1. The convention (axis order): X-Y-Z, Z-Y-X, Z-Y-Z, …
2. Extrinsic (about the fixed spatial axes) or intrinsic (about the moving body axes).

Gimbal lock

For ${\theta }_y=\pi /2$ the first and third rotation axes align. The product $R_x{R}_y{R}_z$ then depends only on ${\theta }_x+{\theta }_z$: a planar rotation is left and one degree of freedom is lost.

Drawbacks: gimbal lock, the parameterization is not unique, lots of conventions. Interpolating Euler angles gives strange paths and non-uniform speed.

Appears in

• Lecture 2.2, Euler Angles: definition, conventions, intrinsic example
• Lecture 2.2, Gimbal lock: derivation for ${\theta }_y=\pi /2$
• Lecture 2.2, Which representation

Related concepts

• Rotation Matrix
• Quaternion