Principal body frame

A principal body frame is an orthonormal basis fixed in the body and aligned with the principal axes of its inertia tensor. In this frame the inertia tensor is diagonal with principal moments $I_1,I_2,I_3$.

A principal axis is an eigenvector direction of the inertia tensor. Choosing three orthogonal principal axes diagonalizes the inertia tensor.

If $e_i$ is fixed in a rigid body with angular velocity $\omega$, then its derivative in the fixed space frame is

If a vector $k$ is fixed in space while a body frame rotates with angular velocity $\omega$, its body-frame components satisfy