Quaternions¶
Quaternions provide a singularity-free representation of 3D rotations.
Overview¶
A quaternion is a four-element mathematical object that can represent any 3D rotation without singularities. In Brahe, quaternions use the scalar-first convention: [w, x, y, z].
Mathematical Representation¶
A unit quaternion is defined as:
\[q = [w, x, y, z]\]
where \(w^2 + x^2 + y^2 + z^2 = 1\)
Advantages¶
- No singularities: Unlike Euler angles, quaternions work for all orientations
- Compact: Only 4 parameters (vs 9 for rotation matrices)
- Efficient: Quaternion multiplication is faster than matrix multiplication
- Interpolation: SLERP (Spherical Linear Interpolation) provides smooth attitude interpolation
Operations¶
Common quaternion operations include:
- Quaternion multiplication (composition of rotations)
- Quaternion conjugate (inverse rotation)
- Quaternion normalization
- Vector rotation