Abstract
This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the Sun to visit the Kuiper belt.
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Broucke R.A.: Solution of the elliptic rendezvous problem with the time as independent variable. J. Guid. Control Dyn. 26(4), 615–621 (2003)
Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (674) (1960)
Cushmann R.H., Bates L.M.: Global Aspects of Classical Integrable Systems. Birkhauser Verlag, Basel (1997)
Escoubet, C.P.: Cluster-II: scientific objectives and data dissemination. Technical report, ESA, ESTEC, Noordwijk, The Netherlands (2000)
Fasano G., D’Errico M.: Modelling orbital relative motion to enable formation design from application requirements. Celest. Mech. Dyn. Astron. 105, 113–139 (2009)
Folta, D., Bristow, J., Hawkins, A., Dell, G.: Enhanced formation flying validation report (gsfc algorithm). Technical report, NASA/GSFC, Maryland, USA (2002)
Gómez G., Marcote M.: High-order analytical solutions of Hill’s equations. Celest. Mech. Dyn. Astron. 94, 197–211 (2006)
Guinn, J.R.: Enhanced formation flying validation report (JPL Algorithm). Technical report, NASA/GSFC, Maryland, USA (2002)
Halsall M., Palmer P.L.: Modelling natural formations of LEO satellites. Celest. Mech. Dyn. Astron. 99, 105–127 (2007)
Hill, G.W.: Researches in the lunar theory. Am. J. Math. 1, 5–26 (129–147, 245–260) (1878)
Imre E., Palmer P.L.: High precission symplectic numerical relative orbit propagation. J. Guid. Control Dyn. 4, 965–973 (2007)
Karlgaard C.D., Lutze F.H.: Second-order relative motion equations. J. Guid. Control Dyn. 26(1), 41–49 (2003)
Lawden D.F.: Fundamentals of space navigation. J. Britsh Interplanet. Soc. 13, 87–101 (1954)
Marsden J.E., Ratiu T., Raugel G.: Symplectic connections and the linearisation of Hamiltonian systems. Proc. Royal Soc. Edinb. 117A, 329–380 (1991)
Melton R.G.: Time-explicit representation of relative motion between elliptical orbits. J. Guid. Control Dyn. 23(4), 604–610 (2000)
Palmer P.L., Imre E.: Relative motion of satellites on neighbouring Keplerian orbits. J. Guid. Control Dyn. 30(2), 521–528 (2007)
Schaub H., Alfriend K.T.: J 2 invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79, 77–95 (2001)
Schweighart S.A., Sedwick R.J.: High-fidelity linearized J 2 model for satellite formation flight. J. Guid. Control Dyn. 25(6), 1073–1080 (2002)
Sengupta P., Vadali S.R., Alfriend K.T.: Second-order state transition for relative motion near perturbed, elliptic orbits. Celest. Mech. Dyn. Astron. 97, 101–129 (2007)
Tapley B.D., Bettadpur S., Watkins M., Reigber C.: The gravity recovery and climate experiment: mission overview and early results. Geophys. Res. Lett. 31(9), L09607 (2004)
Tempesta, P., Winternitz, P., Harnad, J., Miller, Jr. W., Pogosyan, G., Rodriguez, M.: CRM, Proceedings & Lecture Notes, volume 34. Centre de Recherches Mathématiques, Université de Montréal (2004)
Tschauner J., Hempel P.: Optimale beschleunigeungsprogramme fur das rendezvous-manover. Acta Astronautica. 10, 296–307 (1964)
Vaddi S.S., Vadali S.R., Alfriend K.T.: Formation flying: accommodating nonlinearity and eccentricity perturbations. J. Guid. Control Dyn. 26(2), 214–223 (2003)
Wiesel W.E., Pohlen D.J.: Canonical Floquet theory. Celest. Mech. Dyn. Astron. 58(1), 81–96 (1994)
Wiesel W.E.: Relative satellite motion about an oblate planet. J. Guid. Control Dyn. 25(4), 776–785 (2002)
Xiang, W., Jørgensen, J.L.: Formation flying: a subject being fast unfolding in space. In: 5th IAA Symposium on Small Satellites for Earth Observation, Berlin, IAA-B5-0309P (2005).
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Kristiansen, K.U., Palmer, P.L. & Roberts, M. Relative motion of satellites exploiting the super-integrability of Kepler’s problem. Celest Mech Dyn Astr 106, 371–390 (2010). https://doi.org/10.1007/s10569-009-9253-y
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DOI: https://doi.org/10.1007/s10569-009-9253-y