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Euler Angles and Quaternions

Euler angles describe an orientation with three successive rotations, but can suffer from gimbal lock: two rotation axes line up and one degree of freedom is lost. Quaternions use four numbers instead of three, avoid this singularity entirely, and give smoother interpolation between two orientations.

Euler: 3 angles, can gimbal lock\text{Euler: 3 angles, can gimbal lock}limitation of Euler angles
Quaternion: 4 numbers, no gimbal lock\text{Quaternion: 4 numbers, no gimbal lock}advantage of quaternions

Symbols

qqquaternion (4 numbers)

Example

A robot arm's wrist approaches an Euler singularity:

A quaternion avoids the problem entirely.

Gimbal lock is a real singularity in Euler angles, not just a theoretical curiosity.
Practise rotations and transformations for free →

← Homogeneous Transformation · Forward Kinematics →

Part of Robotics: Rotations and transformations.