# Spacecraft-fixed coordinate system

I am working with data from a polar-orbiting satellite (inclination 97 degrees) with limited attitude information. I am trying to define a coordinate system which is fixed with respect to the spacecraft. The attitude control system is designed to keep one axis of the satellite within 0.01 degrees of the geodetic vertical, so I use this as one of the basis vectors (call it $$\hat{e}$$). For the second basis vector, I simply use the component of the satellite's velocity vector orthogonal to $$\hat{e}$$. The third basis vector completes the right handed set. So in summary, my spacecraft-fixed basis vectors are: \begin{align} \hat{s}_1 &= \hat{s}_2 \times \hat{e} \\ \hat{s}_2 &= (\hat{e} \times \mathbf{v}) / | \hat{e} \times \mathbf{v} | \\ \hat{s}_3 &= \hat{e} \end{align}

I represent all vectors above in ECI coordinates.

Now, the satellite has a fluxgate magnetometer onboard, so I can test the quality of the above coordinate system by rotating the fluxgate measurement into an Earth-fixed geographic system and comparing with a reference magnetic model of the Earth.

When I do this, what I find is that the vertical component ($$\hat{e}$$) matches well with the reference model, but there appears to be a slow rotation of the satellite with respect to its velocity vector (the fluxgate horizontal vector must be rotated by about 4 degrees at the equator compared with the pole). In other words, I have to apply a rotation matrix to ($$\hat{s}_1,\hat{s}_2$$) to get the fluxgate measurement to align.

Does anyone know why the satellite would be slowly rotating about the vertical direction along its orbit? Does it have something to do with the Earth's rotation? How can I fix my ($$\hat{s}_1,\hat{s}_2$$) vectors to account for this slow rotation (4 degrees between the equator and pole)?

• have you tried drawing 3 diagrams of your spacecraft in the ($\hat{s}_1,\hat{s}_2$) plane, including lines of longitude and magnetic flux, at seen at both poles and the equator? – JCRM Oct 24 '18 at 15:01
• I'm not familliar with the details, but I wonder if it could be related to different definitions of "vertical"? See answers to In “spacecraft talk” is nadir just a fancy word for “down”? – uhoh Oct 25 '18 at 16:47