Euler Angles¶

Euler angles represent rotations as three sequential rotations about coordinate axes.

Overview¶

Euler angles describe orientation using three angles representing sequential rotations about specified axes. Brahe supports all 12 possible Euler angle sequences (e.g., XYZ, ZYX, ZYZ).

Mathematical Representation¶

An Euler angle rotation is specified by:

Three angles: \((\alpha, \beta, \gamma)\)
A rotation sequence (e.g., XYZ, ZYX)

Common Sequences¶

ZYX (Yaw-Pitch-Roll): Common in aerospace applications
XYZ: Common in robotics
ZYZ: Common in classical mechanics

Advantages¶

Intuitive: Easy to visualize and understand
Minimal: Only 3 parameters
Human-readable: Natural for manual input

Disadvantages¶

Gimbal lock: Singularities occur when middle rotation is ±90°
Ambiguous: Multiple angle sets can represent same orientation
Interpolation: Non-linear, difficult to interpolate smoothly

Gimbal Lock¶

Gimbal lock occurs when the middle rotation approaches ±90 degrees, causing loss of one degree of freedom. Use quaternions to avoid this issue.