Vishal Ray, University of Colorado Boulder; Daniel Scheeres, University of Colorado Boulder
Keywords: Performance metric, Fourier model, drag-coefficient, observability, orbit determination, prediction
Abstract:
Models of non-conservative forces such as atmospheric drag and solar radiation pressure (SRP) for orbit determination and prediction of satellites usually involve the estimation of some force parameter as part of the orbit-fitting process. For the cannonball model, the drag and SRP coefficients are estimated as constants while for Fourier models, the Fourier coefficients are estimated as constants. The performance of higher-fidelity models such as the Fourier models is usually evaluated based on the accuracy of the orbit fit and the improvements in the orbit prediction into the future. The common performance metrics used for this purpose are 1. post-fit residuals, i.e., the error between the actual measurements and filter-predicted measurements, 2. estimation error in initial states, i.e., the error between the estimated initial states and true initial states and 3. error in the estimated parameter such as the effective drag-coefficient. Does an improvement in all the stated metrics for a given model always imply better prediction performance? Through this work, we demonstrate that an improved orbit determination as evaluated using the given performance metrics does not necessarily lead to an improved orbit prediction even in the absence of any other unmodeled dynamics. We introduce a new performance metric – the integrated acceleration error that is more strongly correlated with the orbit prediction than any of the other metrics. This study is carried out in the context of Fourier drag-coefficient models [1,2] but the concepts can be generalized to any other force model in orbit determination and prediction. The performance metrics of the previously proposed body-fixed Fourier (BFF), orbit-fixed Fourier (OFF), body-orbit summation (BOS) and body-orbit double Fourier (BODF) models are computed for different satellite attitude profiles and atmospheric conditions. We show that even if a model has a better post-fit residual, effective drag-coefficient and initial state estimate than the other models, the orbit prediction performance may actually be worse due to a large integrated acceleration error, which for the drag force, reduces to a sum of the drag-coefficient error weighted by the time-varying density and relative velocity squared.
A related aspect to performance evaluation is the observability of estimated parameters. Sometimes the presence of a large number of parameters to be estimated in a force model can lead to a better orbit fit and prediction if there are multiple source of unmodeled dynamics, even if all the parameters are not observable. This occurs due to aliasing of different unmodeled forces, i.e., the filter mis-attributes an unmodeled force to the parameters of a different force being estimated [3]. This may lead to better short-term predictions, but the performance can degrade with longer prediction intervals. Therefore, carrying out an observability analysis is intimately tied to evaluating the performance of any model. We analyze the observability of all the Fourier models using comparative metrics such as rank and singular values of the observability Gramian [4], change in covariance of the estimated coefficients and sensitivity of measurements to the Fourier coefficients. We show that multiple observable subsets are possible for a Fourier model of a given order. The change in performance metrics between different observable subsets is analyzed to show the relative advantages of one subset over another. Ultimately, evaluation of the observability and performance metrics for the Fourier drag models are tied to the orbit determination and prediction of LEO satellites.
[1] Ray, V., Scheeres, D. J., Hesar, S. G., and Duncan, M., A drag coefficient modeling approach using spatial and temporal Fourier expansions for orbit determination, Journal of the Astronautical Sciences, 2019. URL https://doi.org/10.1007/s40295- 019-00200-4.
[2] Ray, V. and Scheeres, D.J., A drag coefficient model to track variations due to attitude and orbital motion, AAS/AIAA Astrodynamics Specialist Conference, Maine, Portland, Aug 11-15 2019. AAS 19-754.
[3] Ray, V. and Scheeres, D.J., Aliasing of unmodeled gravity effects in estimates of non-conservative force coefficients, Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS), Maui, Hawaii, Sep 17-20.
[4] Friedman, Alex M. and Frueh, Carolin, Determining characteristics of artificial near-Earth objects using observability analysis, Acta Astronautica, 144 (2018) 405421. URL https://doi.org/10.1016/j.actaastro.2017.12.028.
Date of Conference: September 15-18, 2020
Track: Astrodynamics