Bounded motion design in the Earth zonal problem using differential algebra based normal form methods

Abstract

Establishing long-term relative bounded motion between orbits in perturbed dynamics is a key challenge in astrodynamics to enable cluster flight with minimum propellant expenditure. In this work, we present an approach that allows for the design of long-term relative bounded motion considering a zonal gravitational model. Entire sets of orbits are obtained via high-order Taylor expansions of Poincarè return maps about reference fixed points. The high-order normal form algorithm is used to determine a change in expansion variables of the map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action–angle coordinates. The action–angle representation of the normal form coordinates is then used to parameterize the original Poincarè return map and average it over a full phase space revolution by a path integral along the angle parameterization. As a result, the averaged nodal period and drift in the ascending node are obtained, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers, and valid for decades.

Appendix

See Tables 8, 9 and 10.

Table 8 Used values for coefficients $J_2$ to $J_{15}$

Table 9 Expansion of $\mathcal {H}_{z}(\delta r,\delta v_r)$, $E(\delta r,\delta v_r)$ and $\omega _p(\delta r,\delta v_r)$ for relative bounded motion orbits with an average nodal period $\overline{T_{d}}=7.64916169$ and an average node drift of $\overline{\Delta \Omega }=1.22871195\text {E-3}$. The expansion is relative to the pseudo-circular LEO from He et al. (2018)

Table 10 Expansion of $\mathcal {H}_{z}(\delta r,\delta v_r)$, $E(\delta r,\delta v_r)$ and $\omega _p(\delta r,\delta v_r)$ for relative bounded motion MEOs with an average nodal period of $\overline{T_{d}}=53.5395648$ and an average node drift of $\overline{\Delta \Omega }=-3.35410945\text {E-4}$. The expansion is relative to the pseudo-circular MEO from Baresi and Scheeres (2017b)