Abstract
A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy–Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4–0.45 in eccentricity and 40–45\(^\circ \) in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy–Wiltshire solution in curvilinear coordinates is also presented.
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Notes
The function \({\mathrm {atan2}}^{*}={\mathrm {mod}}\left( {\mathrm {atan2}} \left( x,y\right) +2\pi ,2\pi \right) \) is here employed in order to return a value in \({[}0;\, 2\pi \)).
In LEO, the J2 differential acceleration is negligible only for very small relative distances, in which case nonlinear gravitational effects would also be negligible and the use of the proposed solution, in place of its linear counterpart, would serve no purpose.
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Acknowledgments
This work has been supported by the Spanish Ministry of Economy and Competitiveness within the framework of the research project “Dynamical Analysis, Advanced Orbital Propagation, and Simulation of Complex Space Systems” (ESP2013-41634-P). The authors also want to thank the funding received from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 607457 (LEOSWEEP). JLG thanks the Spanish Ministry of Education, Culture and Sport for his doctoral fellowship (FPU13/05910). JR thanks “La Caixa” for his doctoral fellowship. The authors are indebted to Javier Hernando-Ayuso and Emilio José Calero-Rodriguez for carefully revising the manuscript and thank the two anonimous reviewers for their valuable suggestions.
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Bombardelli, C., Gonzalo, J.L. & Roa, J. Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates. Celest Mech Dyn Astr 127, 49–66 (2017). https://doi.org/10.1007/s10569-016-9716-x
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DOI: https://doi.org/10.1007/s10569-016-9716-x