Unified Relative Orbital Element Framework for Eccentric Formation Flying
Abstract
Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial--tangential--normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as e→0 and whose first-order geometry is resolved in the velocity--normal--binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal--normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-J2 STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.
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BibTeX
@misc{takubo2026unified,
title = {{Unified Relative Orbital Element Framework for Eccentric Formation Flying}},
author = {Yuji Takubo and Simone D'Amico},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09379},
eprint = {2610.09379},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09379},
}
RIS
TY - UNPB
TI - Unified Relative Orbital Element Framework for Eccentric Formation Flying
AU - Yuji Takubo
AU - Simone D'Amico
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09379
UR - https://arxiv.org/abs/2610.09379
AB - Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial--tangential--normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as e→0 and whose first-order geometry is resolved in the velocity--normal--binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal--normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-J_2 STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.
ER -