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Second-order state transition for relative motion near perturbed, elliptic orbits

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Abstract

This paper develops a tensor and its inverse, for the analytical propagation of the position and velocity of a satellite, with respect to another, in an eccentric orbit. The tensor is useful for relative motion analysis where the separation distance between the two satellites is large. The use of nonsingular elements in the formulation ensures uniform validity even when the reference orbit is circular. Furthermore, when coupled with state transition matrices from existing works that account for perturbations due to Earth oblateness effects, its use can very accurately propagate relative states when oblateness effects and second-order nonlinearities from the differential gravitational field are of the same order of magnitude. The effectiveness of the tensor is illustrated with various examples.

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References

  • Alfriend, K.T., Schaub, H., Gim, D.-W.: Gravitational perturbations, nonlinearity and circular orbit assumption effects on formation flying control strategies. Advan. Astronaut. Sci. 104, 139–158 (2000), also Paper AAS-00-012 of the AAS Guidance and Control Conference

    Google Scholar 

  • Alfriend, K.T., Yan, H., Vadali, S.R.: Nonlinear considerations in satellite formation flying. In: 2002 AIAA/AAS Astrodynamics Specialist Conference, AIAA-2002-4741, Monterey, CA (2002)

  • Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., Reston, VA, revised edition (1999)

  • Bond, V.R.: A new solution for the rendezvous problem. Advan. Astronaut. Sci. 102(2): 1115–1143 (1999), also Paper AAS 99-178 of the AAS/AIAA Space Flight Mechanics Meeting

  • Broucke, R.A. (2003) Solution of the elliptic rendezvous problem with the time as independent variable. J. Guid. Control Dynam. 26(4): 615–621

    Google Scholar 

  • Broucke R.A., Cefola P.J. (1972) On the equinoctial orbit elements. Celest. Mech. 5, 303–310

    Article  ADS  MATH  Google Scholar 

  • Brouwer D. (1959) Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378–397

    Article  ADS  MathSciNet  Google Scholar 

  • Carter T.E. (1990) New form for the optimal rendezvous equations near a Keplerian orbit. J. Guid. Control Dynam. 13(1): 183–186

    MATH  ADS  Google Scholar 

  • Carter T.E. (1998) State transition matrices for terminal rendezvous studies: brief survey and new examples. J. Guid. Control Dynam. 21(1): 148–155

    MATH  Google Scholar 

  • Carter T.E., Humi M. (1987) Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dynam. 10(6): 567–573

    MATH  ADS  Google Scholar 

  • Clohessy W.H., Wiltshire R.S. (1960) Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27, 653–658, 674

    Google Scholar 

  • Euler E.A., Shulman Y. (1967) Second-order solution to the elliptic rendezvous problem. AIAA J. 5(5), 1033–1035

    MATH  ADS  Google Scholar 

  • Feagin T., Gottlieb R.G. (1971) Generalization of Lagrange’s implicit function theorem to n-dimensions. Celest. Mech. 3, 227–231

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Garrison, J.L., Gardner, T.G., Axelrad, P.: Relative motion in highly elliptical orbits. Advan. Astronaut. Sci. 89(2), 1359–1376 (1995), also Paper AAS 95-194 of the AAS/AIAA Space Flight Mechanics Meeting

    Google Scholar 

  • Gim D.-W., Alfriend K.T. (2003) State transition matrix of relative motion for the perturbed noncircular reference. J. Guid. Control Dynam. 26(6): 956–971

    Google Scholar 

  • Gim D.-W., Alfriend K.T. (2005) Satellite relative motion using differential equinoctial elements. Celest. Mech. Dynam. Astron. 92(4): 295–336

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Gurfil P. (2005a) Euler parameters as nonsingular orbital elements in near-equatorial orbits. J. Guid. Control Dynam. 28(5): 1079–1083

    Google Scholar 

  • Gurfil P. (2005b) Relative motion between elliptic orbits: generalized boundedness conditions and optimal formationkeeping. J. Guid. Control Dynam. 28(4): 761–767

    Google Scholar 

  • Hill G.W. (1878) Researches in the lunar theory. Am. J. Math. 1(1): 5–26

    Article  Google Scholar 

  • Junkins J.L., Turner J.D. (1986) Optimal Spacecraft Rotational Maneuvers, Studies in Astronautics, vol 3. Elsevier Science Publishers B. V., The Netherlands

    Google Scholar 

  • Karlgaard C.D., Lutze F.H. (2003) Second-order relative motion equations. J. Guid. Control Dynam. 26(1): 41–49

    Google Scholar 

  • Kasdin N.J., Gurfil P., Kolemen E. (2005) Canonical modelling of relative spacecraft motion via epicyclic orbital elements. Celest. Mech. Dynam. Astron. 92(4): 337–370

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Kaula W.M. (2000) Theory of Satellite Geodesy. Dover Publications, Inc., Mineola, NY

    MATH  Google Scholar 

  • Kechichian J.A. (1998) Motion in general elliptic orbit with respect to a dragging and precessing coordinate frame. J. Astronaut. Sci. 46(1): 25–46

    MathSciNet  Google Scholar 

  • Lawden D.F. (1963) Optimal Trajectories for Space Navigation. Butterworths, London, UK, 2nd edition

    MATH  Google Scholar 

  • Melton R.G. (2000) Time explicit representation of relative motion between elliptical orbits. J. Guid. Control Dynam. 23(4): 604–610

    Google Scholar 

  • Richardson D.L., Mitchell J.W. (2003) A third-order analytical solution for relative motion with a circular reference orbit. J. Astronaut. Sci. 51(1): 1–12

    MathSciNet  Google Scholar 

  • Schweigart S.A., Sedwick R.J. (2002) High-fidelity linearized J 2 model for satellite formation flight. J. Guid. Control Dynam. 25(6): 1073–1080

    Google Scholar 

  • Sengupta P., Sharma R., Vadali S.R. (2006) Periodic relative motion near a general Keplerian orbit with nonlinear differential gravity. J. Guid. Control Dynam. 29(5): 1110–1121

    Article  Google Scholar 

  • Sengupta P., Vadali S.R. (2005) Satellite orbital transfer and formation reconfiguration via an attitude control analogy. J. Guid. Control Dynam. 28(6): 1200–1209

    Google Scholar 

  • Sengupta P., Vadali S.R., Alfriend K.T. (2004) Modeling and control of satellite formations in high eccentricity orbits. J. Astronaut. Sci. 52(1–2): 149–168

    MathSciNet  Google Scholar 

  • Tschauner J.F.A., Hempel P.R. (1965) Rendezvous zu einemin elliptischer bahn umlaufenden ziel. Astronaut. Acta 11(2): 104–109

    Google Scholar 

  • Turner J.D. (2003) Automated generation of high-order partial derivate models. AIAA J. 41(8): 1590–1598

    Article  ADS  Google Scholar 

  • Vadali, S.R.: An analytical solution for relative motion of satellites. In: 5th Dynamics and Control of Systems and Structures in Space Conference, Cranfield University, Cranfield, UK (2002)

  • Vadali S.R., Vaddi S.S., Alfriend K.T. (2002) An intelligent control concept for formation flying satellites. Int. J. Robust Nonlinear Contr. 12, 97–115

    Article  MATH  Google Scholar 

  • Vaddi S.S., Vadali S.R., Alfriend K.T. (2003) Formation flying: accommodating nonlinearity and eccentricity perturbations. J. Guid. Control Dynam. 26(2): 214–223

    Google Scholar 

  • Wolfsberger W., Weiß J., Rangnitt D. (1983) Strategies and schemes for rendezvous on geostationary transfer orbit. Acta Astronaut. 10(8): 527–538

    Article  Google Scholar 

  • Yamanaka K., Ankersen F. (2002) New state transition matrix for relative motion on an arbitrary elliptical orbit. J. Guid. Control Dynam. 25(1): 60–66

    Article  Google Scholar 

  • Yan, H.: Dynamics and Real-Time Optimal Control of Aerospace Systems. Dissertation, Texas A&M University, College Station, TX (2006)

  • Yan, H., Sengupta, P., Vadali, S.R., Alfriend, K.T.: Development of a state transition matrix for relative motion using the unit sphere approach. Advan. Astronaut. Sci. 119(1), 935–946 (2004), also Paper AAS 04-163 of the AAS/AIAA Space Flight Mechanics Meeting

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Authors and Affiliations

  1. Department of Aerospace Engineering, TAMU 3141, Texas A&M University, College Station, TX, 77843-3141, USA

    Prasenjit Sengupta, Srinivas R. Vadali & Kyle T. Alfriend

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  1. Prasenjit Sengupta
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  2. Srinivas R. Vadali
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  3. Kyle T. Alfriend
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Correspondence to Prasenjit Sengupta.

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Sengupta, P., Vadali, S.R. & Alfriend, K.T. Second-order state transition for relative motion near perturbed, elliptic orbits. Celestial Mech Dyn Astr 97, 101–129 (2007). https://doi.org/10.1007/s10569-006-9054-5

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  • DOI: https://doi.org/10.1007/s10569-006-9054-5

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