# Covariance frame transformation with orbital and propagation parameters

Hello the community,

I’m currently working on a solution to perform frame transformations on a 9x9 CDM-standard covariance matrix including both the orbital parameters (position, velocities) and propagation parameters (solar radiation pressure, atmospheric drag and thrust acceleration).
In my understanding, the additional propagation parameters act as perturbative forces that could be associated to acceleration effects impacting the satellite orbital state.

Knowing that, I’m considering an Orekit-based frame transformation model where PVT derivatives are implemented to compute the jacobian transformation matrix.

`````` // Input 9x9 cartesian covariance matrix
RealMatrix cartesianCovarianceIn = ... ;

// Define the frame transform
Transform inToOut = frameIn.getTransformTo(frameOut, orbit.getDate());

// Compute the jacobian matrix
double[][] jacobian = new double[9][9];
inToOut.getJacobian(CartesianDerivativesFilter.USE_PVT, jacobian);
RealMatrix j = new Array2DRowRealMatrix(jacobian, false);

// Perform the covariance frame transformation
RealMatrix cartesianCovarianceOut = j.multiply(cartesianCovarianceIn.multiplyTransposed(j));
``````

I turn back to you for some feedback on the initial assumptions and built model. Unfortunatly I have limited access to reference litterature on the topic. The Vallado reference paper on covariance transformation is hardly accessible and the link shared at the time in this thread has expired…

I thank you in advance for your time and assistance.

Best regards,
Gueorguy

I would be generally careful. The covariances associated with position and velocity are of a completely different order than drag or srp, particularly for a high altitude LEO (low altitude may differ for drag, but not for srp).

In my experience (with UKF), the covariance propagation of these terms simply does not generate a significant difference for position and velocity over a short timeframe (say, 60 seconds). Therefore, the drag and srp parameters cannot be estimated properly with this large state matrix.

Hi @egemenimre,

Thank you for your feedback. Indeed the unit and orders of a covariance matrix including both orbital and propagation parameters are not uniform.
The orbital-related sigmas are position and velocity-type units, where the CR, SRP and THR related sigmas of propagation parameters (see CCSDS covariance format) are of inertial moment or acceleration types.
My objective is to establish a frame transformation metholology of an input covariance with already estimated orbital and propagation parameters.

Hi @Gueorguy,

I hope I understand your question well.
In my opinion, you just need to add an identity matrix of the size of your propagation parameters vector (so I_3 in your case I guess) to the Jacobian J in your first post.
Then you can apply the formula you already used: P_{out} = J * P_{in} * J^T

Hope this helps,
Maxime

Hi @MaximeJ,

Thanks for your feedback.
Do you mean adding a 3x3 identity matrix to a 6x6 jacobian based on position-velocity derivaties and thus discarding the acceleration derivatives ?

Regards,
Gueorguy

Hi @Gueorguy,

Yes, that’s what I meant. To be honest I missed the fact that you were using the acceleration derivatives in your first Jacobian.
I’ve never seen any code using these. And I don’t think the CCSDS CDM-Standard can handle them. I’ve checked quickly and the last orbital 6x6 part of the covariance is the velocity on LOF Z-axis, and then the format switches to the propagation parameters.

Regards,
Maxime

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Hi @MaximeJ,

It does make sense indeed. In addition to that, I’ve rarely seen CDM reports with fully 9x9 populated covariances matrices. It’s quiet tiedious to find reference use cases on the matter.
However the fact that we are dealing with acceleration-type derivatives (thrust acceleration) in one of the three propagation parameters, did raise my concern on the matter of frame transformation.

Btw, can I ask you if you still have access to the Vallado, Covariance Transformations for Satellite Flight Dynamics Operations paper @bcazabonne shared at the time in a previous thread ?

Best regards,
Gueorguy

No sorry it looks like it’s publicly available anymore

Maxime

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Covariance Transformations.pdf (333.8 KB)

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Hi,
For those interested in the matter of covariance propagation of propagation parameters (SRP, Drag, Cross-Track acceleration), I’ve found this well detailed ESOC paper on covariance propagation based on a linear stochastic parameters method integrating time-correlated errors evolution within the estimated propagation parameters.

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