Fast Dynamic Orbit Determination of LEO Satellites using the Legendre Polynomial Approximation

The Low Earth Orbiting (LEO) satellites are widely used for geosciences applications. For most applications, precise orbital information of the satellites is required. A Combination of the in suite observations and dynamic orbit yields the optimum solution. In order to obtain the combined optimal solution, one needs to analytically or numerically propagate the state vector epoch by an epoch based on dynamic force models. In the analytical propagation, the dynamic model simplification leads to a biased solution. On the other hand, the numerical solution is a highly time- consuming computational task. Among all computation parts, computation of the Legendre Polynomials is the most time- consuming. Vectorization could effectively reduce computation time of the polynomials for the gravity field modeling. However, it cannot be implemented for the orbit determination because of the point-to-point computation in orbit propagation.
In this article, we propose a new method for effective computation of the Legendre polynomials. The proposed method is based on the approximation of the polynomials at any arbitrary point using pre-computed values of the polynomials on evenly-spaced grid points (i.e., mesh points). Moreover, the first- and second-order derivatives of the polynomials are simultaneously computed using the recurrence relations at the mesh points. Therefore, the polynomials can be approximated using the Hermite approximation algorithm between the mesh points. In other words, employing the Legendre polynomial derivatives for estimating the best approximating function efficiently prevents the approximating oscillations of polynomials between the mesh points. Consequently, the approximating function perfectly follows the Legendre polynomials between the mesh points.
Of course, a few methods have been proposed for fast computation of dynamic orbits by other researchers. They are mostly based on the computation of the gravitational acceleration on the mesh points and approximation of the acceleration between the mesh points. Our proposed method leads to higher accuracy since both the Legendre polynomials and their derivatives are used for approximation.
In order to show numerical performance, the proposed method has been implemented for a CHAMP-type LEO satellite orbit propagation. Sub-millimeter approximation error can be achieved for a two-week propagated orbit.