● Domain 04 · Space Sciences & Technology
It stays a rotation.
Attitude does not live in a vector space. It lives on a group, and an estimator that forgets this returns objects that are not rotations.
Satellite constellationTwenty four spacecraft on three inclined planes. The view closes on one as it slews to a new attitude on its wheels and thrusters.Constellation geometry and attitude control · Walker, J. Br. Interplanet. Soc. 37 (1984); Wertz, Spacecraft Attitude Determination and Control (1978)
Repairing the symptom
Integrate nine unconstrained numbers and the matrix stops being orthogonal. It still multiplies, still looks like an answer, and everything downstream inherits the distortion. Renormalising afterwards takes the symptom away and leaves the cause in place.
What changes
Confine the state to the group and the estimate is a rotation at every step, not a nearby object repaired into one. Orthogonality is not maintained. It is not available to lose.
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 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.
Sources
RᵀR = I, det R = 1
The definition of the rotation group · Murray, Li & Sastry, A Mathematical Introduction to Robotic Manipulation (1994)
S3 → SO(3), a double cover
Unit quaternions and attitude · Shuster, J. Astronaut. Sci. 41 (1993)
Is this your problem?
Tell us what has to hold, and what happens if it does not.
xzenoverseindustries@gmail.com · answered within two business days