Demonstration
One angular velocity, two integrations.
First order update
R ← R + h[ω]×R, the same thing to first order, and what a model that treats an attitude as nine free numbers will learn. It leaves the group on the first step.
Exact exponential
R ← R exp([ω]×h), through Rodrigues' formula. The exponential of a skew matrix is orthogonal identically, so the frame stays on the group with nothing to reproject.
The state is confined to the rotation group, so the estimate stays a rotation as it is stepped.
How to read this figure
The body, built from the three frame axes, so a shear shows in the shape.
The three columns of the matrix: the body axes. On the group, unit length and at right angles.
The unit sphere the tip of every axis is confined to while the frame is still a rotation.
An axis that is no longer unit length, and the distance from the group under each figure.
The state is confined to the rotation group, so the estimate stays a rotation as it is stepped. Columns orthonormal, determinant one: what is measured is the distance from the group. Both columns integrate the same angular velocity. The first order update reaches a distance above one, with a determinant near 2.2 and a body more than double the volume it started with; the exponential holds the frame on the group to thirteen decimal places, its determinant still reading one.