High Precision Symplectic Numerical Relative Orbit Propagation

Egemen Imre & P L Palmer

This paper presents a numerical method to propagate relative orbits. It can handle
up to an arbitrary number of zonal and tesseral geopotential terms and can be extended
to accommodate the effects of atmospheric drag as well as other perturbations. This
method relies on defining a ‘relative Hamiltonian,’ which describes both the absolute
and the relative motion of two satellites. Exploiting the separability of the solution,
the Keplerian motion is described via analytical means whereas the effects of higher
order terms are handled via a symplectic numerical integration scheme. The derivation
and the numerical integration are designed to conserve the constants of the motion,
resulting in better long term accuracy.
When used within a relative orbit estimator, such a high precision relative orbit
propagator should reduce the frequency of the required sensor input dramatically for
a given estimation accuracy.
We present results for a broad range of scenarios with large separations and show
that sub-metre accuracy is possible over five days of propagation with a geopotential
model containing 36 terms in tesseral and zonal harmonics. These results are valid
for eccentricities reaching 0.5. Furthermore, the relative propagation scheme is significantly
faster than differencing two absolute orbit propagations.

Paper: Numerical Relative Propagation

Ref: Imre, E. & Palmer, P.L., AIAA J Guidance, Control & Dynamics, 30, No 4, pp. 965 — 973,  2007.

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