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Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

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Abstract

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J2-invariant orbits are obtained.

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References

  • K. Alfriend H. Schaub (2000) ArticleTitle‘Dynamics and control of spacecraft formations: challenges and some solutions’ J. Astronaut. Sci. 48 IssueID2 249–267

    Google Scholar 

  • Alfriend, K. and Yan, H.: (2002), ‘An orbital elements approach to the nonlinear formation flying problem’, In: International Formation Flying Symposium, Toulouse, France.

  • Battin, R.: (1999), An Introduction to the Mathematics and Methods of Astrodynamics, AIAA.

  • Born G.H. (2001). ‘Motion of a Satellite under the influence of an Oblate Earth’. Technical Report ASEN 3200. University of Colorado, Boulder.

  • T. Carter M. Humi (1987) ArticleTitle‘Fuel-optimal rendezvous near a point in general Keplerian orbit’ J. Guidance, Control Dynam. 10 IssueID6 567–573

    Google Scholar 

  • W. Clohessy R. Wiltshire (1960) ArticleTitle‘Terminal guidance system for satellite rendezvous’ J. Astronaut. Sci. 27 IssueID9 653–678

    Google Scholar 

  • M. Efroimsky P. Goldreich (2003) ArticleTitle‘Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach’ J. Math. Phys. 44 5958–5977 Occurrence Handle10.1063/1.1622447

    Article  Google Scholar 

  • M. Efroimsky P. Goldreich (2004) ArticleTitle‘Gauge freedom in the N-body problem of celestial mechanics’ Astron. Astrophys. 415 1187–1899 Occurrence Handle10.1051/0004-6361:20034058

    Article  Google Scholar 

  • Gim, D. and Alfriend, K.: (2001), ‘The state transition matrix of relative motion for the perturbed non-circular reference Orbit’, In: Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA.

  • Goldstein, H.: (1980), Classical Mechanics. Addison-Wesley.

  • P. Gurfil N. Kasdin (2004) ArticleTitle‘Nonlinear modeling and control of spacecraft relative motion in the configuration space’ J. Guidance, Control, Dynam. 27 IssueID1 154–157

    Google Scholar 

  • G. Hill (1878) ArticleTitle‘Researches in the lunar theory’ Am. J Math. 1 5–26

    Google Scholar 

  • G. Inalhan M. Tillerson J. How (2002) ArticleTitle‘Relative dynamics and control of spacecraft formations in eccentric orbits’ J. Guidance, Control Dynam. 25 IssueID1 48–60

    Google Scholar 

  • Karlgaard, C. and Lutze, F.: (2001), ‘Second-order relative motion equations’. In: Proceedings of the AAS/AIAA Astrodynamics Conference, Quebec City, Quebec.

  • Kasdin, N. and Gurfil, P.: (2004), ‘Hamiltonian modeling of relative motion’. In: E. Belbruno, D. Folta and P. Gurfil (eds.), Astrodynamics, Space Missions, and Chaos, The Annals of the New York Academy of Sciences, Vol. 1017.

  • F. Namouni (1999) ArticleTitle‘Secular interactions of coorbiting objects’ Icarus 137 293–314 Occurrence Handle10.1006/icar.1998.6032

    Article  Google Scholar 

  • H. Schaub K. Alfriend (2001) ArticleTitle‘J2 invariant relative orbits for spacecraft formations’ Celes. Mech. Dynam. Astron. 79 77–95 Occurrence Handle10.1023/A:1011161811472

    Article  Google Scholar 

  • H. Schaub S. Vadali K. Alfriend (2000) ArticleTitle‘Formation flying control using mean orbital elements’ J. Astronaut. Sci. 48 IssueID1 69–87

    Google Scholar 

  • D. Scheeres F. Hsiao N. Vinh (2003) ArticleTitle‘Stabilizing motion relative to an unstable orbit: Applications to spacecraft formation flight’ J. Guidance, Control Dynam. 26 IssueID1 62–73

    Google Scholar 

  • Vinti J.P. (1998). Orbital and Celestial Mechanics, Progress in Astronautics and Aeronautics, Vol. 177, AIAA.

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Authors and Affiliations

  1. Mechanical and Aerospace Engineering Department, Princeton University, Princeton, NJ, 08544, USA

    N. Jeremy Kasdin & Egemen Kolemen

  2. Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa, 32000, Israel

    Pini Gurfil

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  1. N. Jeremy Kasdin
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  2. Pini Gurfil
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  3. Egemen Kolemen
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Correspondence to N. Jeremy Kasdin.

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Kasdin, N.J., Gurfil, P. & Kolemen, E. Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements. Celestial Mech Dyn Astr 92, 337–370 (2005). https://doi.org/10.1007/s10569-004-6441-7

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  • DOI: https://doi.org/10.1007/s10569-004-6441-7

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