Math & Kinematics

Quaternion

A quaternion is a four-dimensional number whose unit elements represent 3D rotations compactly and without the gimbal-lock singularities of Euler angles. Unit quaternions double-cover the rotation group SO(3), with q and -q encoding the same rotation, and support smooth interpolation via slerp. They are the standard orientation representation in robot state estimation, control, and simulation.

Why it matters for physical AI

Orientation representation choices ripple through estimators, controllers, and learned models; quaternions' compactness and singularity-freedom make them the default, though the double cover demands care in losses and filters.