Analysis of Fidelities of Linearized Orbital Models using Least Squares

S A A Gilani & P L Palmer

 

Satellites orbiting in Low Earth Orbit (LEO) are
accelerated by Earth gravity and dominant orbital
perturbations due to Earth oblateness and atmospheric drag.
The equations of motion describing such a motion are
highly nonlinear in nature. Linearized orbital models only
approximate these nonlinear dynamics. The difference
between a linearized model and full nonlinear dynamical
equations of motion is termed as process noise. To
determine the accuracy of these approximate models, we
need to compare these with numerical integration of the full
non-linear dynamical equations. Linearized solution
propagations are characterized by a set of initial conditions
which determine orbital evolution. The question arises on
how to choose the initial conditions of the analytical
approximation appropriate to a given choice of initial
conditions for the numerically propagated orbit such that the
process noise is minimized. An algorithm is developed,
based upon the statistical method of nonlinear least squares
to compare linearized orbital models for relative and
absolute satellite dynamical motion with numerically
propagated orbits to evaluate their accuracy. Due to recent
interest in formation flying missions a comparison of
accuracies of linearized relative orbital models i.e., Hill-
Clohessy-Wiltshire (HCW) equations, J2 Modified Hills
equations by Schweighart-Sedwick (SS), and for
absolute orbital models described by analytical equations for
Kepler’s problem and Epicycle Model by Hashida and
Palmer have been carried out.

 

Paper: Epicycle Model Comparison

Ref: Gilani, S.A. & Palmer, P.L., IEEE Aerospace Conf, Big Sky, Montana, USA, March 2011.

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