A parameterized model to achieve orbit extrapolation on-board for highly elliptical orbits

Abstract

<p indent="0mm">In general, both numerical and analytical methods can be used to predict the orbit. Numerical integrations are often adopted to achieve high accuracy, while analytical methods are more efficient. Many methods can be successfully used to predict the orbit for near-circular orbits. However, when it comes to orbit extrapolation on-board for orbits of extreme eccentricities, these existing methods are impractical. Conventional numerical integrators are extremely slow and analytical methods cannot reach sufficient accuracy because of the simplified force model. To find an appropriate solution to achieve orbit extrapolation on-board for highly elliptical orbits, a parameterized model comprising a polynomial fitting formula is proposed herein. The exact form of the fitting formula was determined after the analysis of the analytical solution of perturbations. Different functions were used to reflect the periodic term and secular term under the effect of perturbations. Five orbital elements <italic>σ</italic> (true anomaly <italic>f</italic>, argument of perigee<italic> ω</italic>, right ascension of ascending node <italic>Ω</italic>, orbital inclination <italic>i</italic> and geocentric distance <italic>r</italic>) were first fitted and then calculated using the fitting formula. Real positions of satellites were also calculated using these five orbital elements and presented in an inertial coordinate system. A total of 33 Molniya orbits were tested using this method. Twenty-eight groups of tests were performed for each satellite, and each extrapolation spanned three days. The numerical test results indicate that residuals of orbital elements (<italic>f</italic>, <italic>w</italic>, <italic>i</italic> and <italic>r</italic>) are affected by the eccentricity of orbits. Position errors are generally several hundred meters in three days.