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Analytical Derivation of Single-Impulse Maneuvers Guaranteeing Bounded Relative Motion Under J2

Published Online:https://doi.org/10.2514/1.60238

Keeping a cluster of satellites within bounded relative distances requires active control, because natural perturbations (the most significant of which is the J2 term in the geopotential) tend to move the satellites apart. Whereas there is abundant literature on controlling the relative drift using multiple impulsive maneuvers and approximate astrodynamical models involving mean orbital elements, the works that attempt to minimize the number of impulses while using the inertial position and velocity vectors of the satellites are scarce. In this paper, single-impulse distance-keeping maneuvers are derived, without approximating the J2-perturbed dynamics. An analytical derivation of minimum-fuel impulsive maneuvers between equatorial orbits is provided, while relying on radial period and orbital angle matching conditions. Then, a continuation procedure is used to obtain single-impulse relative distance control between inclined orbits. The development of the impulsive maneuvers relies on the inertial position and velocity vectors of the satellites and does not involve mean elements. In each step, necessary and sufficient conditions for the existence of a single-impulse maneuver are provided. The results are illustrated using a number of realistic scenarios, such as a simulation that includes a 21×21 gravitational model, drag, and lunisolar attraction.

References

  • [1] Alfriend K., Vadali S., Gurfil P., How J. and Breger L., Spacecraft Formation Flying: Dynamics, Control and Navigation, Elsevier, Oxford, 2010, pp. 185–222. CrossrefGoogle Scholar

  • [2] Ross I. M., “Linearized Dynamic Equations for Spacecraft Subject to J2 Perturbations,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 4, 2003, pp. 657–659.doi:https://doi.org/10.2514/2.5095 JGCDDT 0162-3192 LinkGoogle Scholar

  • [3] Hamel J. F. and de Lafontaine J., “Linearized Dynamics of Formation Flying Spacecraft on a J2-Perturbed Elliptical Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 6, 2007, pp. 1649–1658.doi:https://doi.org/10.2514/1.29438 JGCDDT 0162-3192 LinkGoogle Scholar

  • [4] Vadali S. R., “An Analytical Solution for Relative Motion of Satellites,” Fifth Dynamics and Control of Systems and Structures in Space Conference, Cranfield Univ., Cranfield, England, U.K., July 2002. Google Scholar

  • [5] Gim D.-W. and Alfriend K. T., “State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 6, 2003, pp. 956–971.doi:https://doi.org/10.2514/2.6924 JGCDDT 0162-3192 LinkGoogle Scholar

  • [6] Gim D.-W. and Alfriend K. T., “Satellite Relative Motion Using Differential Equinoctial Elements,” Celestial Mechanics and Dynamical Astronomy, Vol. 92, No. 4, Aug. 2005, pp. 295–336.doi:https://doi.org/10.1007/s10569-004-1799-0 CrossrefGoogle Scholar

  • [7] Wiesel W., “Relative Satellite Motion About an Oblate Planet,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4, 2002, pp. 776–785.doi:https://doi.org/10.2514/2.4946 JGCDDT 0162-3192 LinkGoogle Scholar

  • [8] Humi M. and Carter T., “Orbits and Relative Motion in the Gravitational Field of an Oblate Body,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 3, 2008, pp. 522–532.doi:https://doi.org/10.2514/1.32413 JGCDDT 0162-3192 LinkGoogle Scholar

  • [9] Martinusi V. and Gurfil P., “Solutions and Periodicity of Satellite Relative Motion Under Even Zonal Harmonics Perturbations,” Celestial Mechanics and Dynamical Astronomy, Vol. 111, No. 4, 2011, pp. 387–414.doi:https://doi.org/10.1007/s10569-011-9376-9 0923-2958 CrossrefGoogle Scholar

  • [10] Lara M. and Gurfil P., “J2 Perturbation Solution to the Relative Motion Problem,” IAA Conference on Dynamics and Control of Space Systems, Paper  IAA-AAS-DyCoSS1-06-02, March 2012. Google Scholar

  • [11] Schaub H. and Alfriend K. T., “Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formations,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, 2001, pp. 739–745.doi:https://doi.org/10.2514/2.4774 JGCDDT 0162-3192 LinkGoogle Scholar

  • [12] D'Amico S., Gill E. and Montenbruck O., “Relative Orbit Control Design for the Prisma Formation Flying Mission,” AIAA Guidance, Navigation and Control Conference and Exhibit, Keystone, CO, AIAA Paper  2006-6067, 2006. Google Scholar

  • [13] Gill E., D’Amico S. and Montenbruck O., “Autonomous Formation Flying for the PRISMA Mission,” Journal of Spacecraft and Rockets, Vol. 44, No. 3, 2007, pp. 671–681.doi:https://doi.org/10.2514/1.23015 JSCRAG 0022-4650 LinkGoogle Scholar

  • [14] Mishne D., “Formation Control of Satellites Subject to Drag Variations and J2 Perturbations,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 4, 2004, pp. 685–692.doi:https://doi.org/10.2514/1.11156 JGCDDT 0162-3192 LinkGoogle Scholar

  • [15] Beigelman I. and Gurfil P., “Optimal Fuel-Balanced Impulsive Formationkeeping for Perturbed Spacecraft Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, 2008, pp. 1266–1283.doi:https://doi.org/10.2514/1.34266 JGCDDT 0162-3192 LinkGoogle Scholar

  • [16] Lovell T. A. and Tragesser S. G., “Guidance for Relative Motion of Low Earth Orbit Spacecraft Based on Relative Orbit Elements,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, RI, AIAA Paper  2004-4988, 2004. Google Scholar

  • [17] Vaddi S. S., Alfriend K. T., Vadali S. R. and Sengupta P., “Formation Establishment and Reconfiguration Using Impulsive Control,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 2, 2005, pp. 262–268.doi:https://doi.org/10.2514/1.6687 JGCDDT 0162-3192 LinkGoogle Scholar

  • [18] Gurfil P., “Relative Motion Between Elliptic Orbits: Generalized Boundedness Conditions and Optimal Formationkeeping,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 4, 2005, pp. 761–767.doi:https://doi.org/10.2514/1.9439 JGCDDT 0162-3192 LinkGoogle Scholar

  • [19] Arnold V. I., Mathematical Aspects of Classical Mechanics, 2nd ed., Springer, Berlin, 1989, pp. 34–36. CrossrefGoogle Scholar

  • [20] Gurfil P., Herscovitz J. and Pariente M., “The SAMSON Project—Cluster Flight and Geolocation with Three Autonomous Nano-Satellites,” 26th AIAA/USU Conference on Small Satellites, AIAA/USU Paper  SSC12-VII-2, Aug. 2012. Google Scholar