Skip to main content
SpringerLink
Log in

Describing relative motion near periodic orbits via local toroidal coordinates

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

Local toroidal coordinate systems are introduced to characterize relative motion near a periodic orbit with an oscillatory mode in the circular restricted three-body problem. These coordinate systems are derived from a first-order approximation of invariant tori relative to a periodic orbit and supply a geometric interpretation that is consistent across distinct periodic orbits. First, the local toroidal coordinate sets are used to rapidly generate first-order approximations of quasi-periodic relative motion. Then, geometric properties of these first-order approximations are used to predict the minimum and maximum separation distances between a spacecraft following quasi-periodic motion relative to another spacecraft located on a periodic orbit. Implementation of the local toroidal coordinate systems and associated geometric analyses is demonstrated in the context of spacecraft formations operating near members of the Earth–Moon \(L_2\) southern halo orbit family.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Availability of data and materials

No additional data or materials are provided.

References

  • Alfriend, K., Vadali, S.R., Gurfil, P., How, J., Breger, L.: Spacecraft Formation Flying: Dynamics, Control and Navigation, vol. 2. Chapter 4: Nonlinear Models of Relative Dynamics, pp. 59–82. Elsevier, Amsterdam (2009)

  • Bakhtiari, M., Daneshjou, K., Abbasali, E.: A new approach to derive a formation flying model in the presence of a perturbing body in inclined elliptical orbit: relative hovering analysis. Astrophys. Space Sci. 362(2), 36 (2017). https://doi.org/10.1007/s10509-016-2968-9

    Article  ADS  MathSciNet  Google Scholar 

  • Barden, B.T., Howell, K.C.: Formation flying in the vicinity of libration point orbits. In: Advances in the Astronautical Sciences, pp. 969–988 (1998a)

  • Barden, B.T., Howell, K.C.: Fundamental motions near collinear libration points and their transitions. J. Astronaut. Sci. 46(4), 361–378 (1998b). https://doi.org/10.1007/BF03546387

    Article  ADS  MathSciNet  Google Scholar 

  • Barden, B.T., Howell, K.C.: Dynamical issues associated with relative configurations of multiple spacecraft near the Sun-Earth/Moon L1 point. In: AAS/AIAA Astrodynamics Specialist Conference (1999)

  • Baresi, N., Scheeres, D.J.: Design of bounded relative trajectories in the Earth zonal problem. J. Guid. Control Dyn. 40(12), 3075–3087 (2017). https://doi.org/10.2514/1.G002603

    Article  ADS  Google Scholar 

  • Baresi, N., Scheeres, D.J., Schaub, H.: Bounded relative orbits about asteroids for formation flying and applications. Acta Astronaut. 123, 364–375 (2016). https://doi.org/10.1016/j.actaastro.2015.12.033

    Article  ADS  Google Scholar 

  • Belvin, W., Doggett, B.R., Watson, J., Dorsey, J.T., Warren, J., Jones, T.C., Komendera, E., Mann, T., Bowman, L.: In-space structural assembly: applications and technology. In: AIAA Spacecraft Structures Conference (2016)

  • Bennett, T., Schaub, H.: Continuous-time modeling and control using nonsingular linearized relative-orbit elements. J. Guid. Control Dyn. 39(12), 2605–2614 (2016). https://doi.org/10.2514/1.G000366

    Article  ADS  Google Scholar 

  • Breakwell, J.V., Brown, J.V.: The halo family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem. Celest. Mech. 20(4), 389–404 (1979)

    Article  ADS  Google Scholar 

  • Burt, J., Smith, B.: Deep space climate observatory: the DSCOVR mission. In: IEEE Aerospace Conference, Big Sky, MT (2012). https://doi.org/10.1109/AERO.2012.6187025

  • Calico, R.A., Wiesel, W.E.: Control of time-periodic systems. J. Guid. Control Dyn. 7(6), 671–676 (1984). https://doi.org/10.2514/3.19911

    Article  ADS  MATH  Google Scholar 

  • Carter, T.E.: State transition matrices for terminal rendezvous studies: Brief survey and new example. J. Guid. Control Dyn. 21(1), 148–155 (1998). https://doi.org/10.2514/2.4211

    Article  ADS  MATH  Google Scholar 

  • Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (1960). https://doi.org/10.2514/8.8704

    Article  MATH  Google Scholar 

  • Crusan, J.C., Smith, R.M., Craig, D.A., Caram, J.M., Guidi, J., Gates, M., Krezel, J.M., Herrmann, N.B.: Deep space gateway concept: extending human presence into cislunar space. In: IEEE Aerospace Conference, pp. 1–10. IEEE (2018)

  • Dunham, D.W., Roberts, C.E.: Stationkeeping techniques for libration-point satellites. J. Astronaut. Sci. 49(1), 127–144 (2001). https://doi.org/10.1007/BF03546340

    Article  ADS  MathSciNet  Google Scholar 

  • Elliott, I., Bosanac, N.: Impulsive control of formations near invariant tori via local toroidal coordinates. In: 32nd AAS/AIAA Astrodynamics Specialist Conference (2021a)

  • Elliott, I., Bosanac, N.: Spacecraft formation control near a periodic orbit using geometric relative coordinates. In: 31st AAS/AIAA Spaceflight Mechanics Meeting (2021b)

  • Elliott, I., Bosanac, N.: Characterizing natural relative motion of formations on invariant tori in multi-body systems. In: AIAA Scitech 2022 Forum (2022)

  • Farquhar, R.W.: The utilization of halo orbits in advanced lunar operations. NASA Technical Note D-6365 (1971)

  • Farres, A., Webster, C., Folta, D.: High fidelity modeling of SRP and its effect on the relative motion of Starshade and WFIRST. In: Space Flight Mechanics Meeting (2018). https://doi.org/10.2514/6.2018-2227

  • Ferrari, F., Lavagna, M.: Suitable configurations for triangular formation flying about collinear libration points under the circular and elliptic restricted three-body problems. Acta Astronaut. 147, 374–382 (2018). https://doi.org/10.1016/j.actaastro.2016.08.011

    Article  ADS  Google Scholar 

  • Franzini, G., Innocenti, M.: Relative motion dynamics in the restricted three-body problem. J. Spacecr. Rocket. 56(5), 1322–1337 (2019). https://doi.org/10.2514/1.A34390

    Article  ADS  Google Scholar 

  • Gardner, J., Mather, J., Clampin, M., Doyon, R., Greenhouse, M., Hammel, H., Hutchings, J., Jakobsen, P., Lilly, S., Long, K., Lunine, J., Mccaughrean, M., Mountain, M., Nella, J., Rieke, G., Rieke, M., Rix, H.-W., Smith, E., Sonneborn, G., Wright, G.: The James Webb space telescope. Space Sci Rev 123, 485–606 (2009). https://doi.org/10.1007/s11214-006-8315-7

    Article  ADS  Google Scholar 

  • Gill, E., D’Amico, S., Montenbruck, O.: Autonomous formation flying for the PRISMA mission. J. Spacecr. Rocket. 44(3), 671–681 (2007). https://doi.org/10.2514/1.23015

    Article  ADS  Google Scholar 

  • Gómez, G., Masdemont, J., Simó, C.: Quasihalo orbits associated with libration points. JAS 46(2), 135–176 (1998)

    MathSciNet  Google Scholar 

  • Gurfil, P., Kasdin, N.: Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem. Acta Astronaut. 54(6), 433–453 (2004)

    Article  ADS  Google Scholar 

  • Hazeltine, R.D., Meiss, J.D.: Plasma Confinement, Chapter 2: Review of Fundamentals, pp. 21–36. Dover Publications, New York (2003)

  • Henry, D.B., Scheeres, D.J.: Expansion maps: designing relative trajectories on quasi-periodic orbits. J. Guid. Control Dyn. 44(3), 457–468 (2021). https://doi.org/10.2514/1.G005492

    Article  ADS  Google Scholar 

  • Héritier, A., Howell, K.C.: Dynamical evolution of natural formations in libration point orbits in a multi-body regime. Acta Astronaut. 102, 332–340 (2014). https://doi.org/10.1016/j.actaastro.2013.10.017

    Article  ADS  Google Scholar 

  • Howell, K.C.: Three-dimensional, periodic, halo orbits. Celest. Mech. 32(1), 53–71 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  • Howell, K.C., Marchand, B.G.: Natural and non-natural spacecraft formations near the L1 and L2 libration points in the Sun-Earth/Moon ephemeris system. Dyn. Syst. 20(1), 149–173 (2005). https://doi.org/10.1080/1468936042000298224

    Article  MathSciNet  MATH  Google Scholar 

  • Hsiao, F., Scheeres, D.: The dynamics of formation flight about a stable trajectory. J. Astronaut. Sci. 50(3), 269–287 (2002). https://doi.org/10.1007/BF03546252

    Article  ADS  MathSciNet  Google Scholar 

  • Jorba, À., Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D Nonlinear Phenom. 132(1), 189–213 (1999). https://doi.org/10.1016/S0167-2789(99)00042-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kasper, J., Lazio, J., Romero-Wolf, A., Lux, J., Neilsen, T.: The Sun Radio Interferometer Space Experiment (SunRISE) Mission Concept. In: 2019 IEEE Aerospace Conference, pp. 1–11 (2019). https://doi.org/10.1109/AERO.2019.8742146

  • Kolemen, E., Kasdin, N., Gurfil, P.: Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem. Celest. Mech. Dyn. Astron. 112, 47–74 (2012). https://doi.org/10.1007/s10569-011-9383-x

    Article  ADS  MATH  Google Scholar 

  • Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Marsden Books (2006)

  • Le Moigne, J.: Distributed spacecraft missions (DSM) technology development at NASA Goddard Space Flight Center. In: Institute of Electrical and Electronics Engineers International Geoscience and Remote Sensing Symposium, pp. 293–296 (2018). https://doi.org/10.1109/IGARSS.2018.8519065

  • Lo, M.W.: Satellite-constellation design. Comput. Sci. Eng. 1, 58–67 (1999). https://doi.org/10.1109/5992.743623

    Article  Google Scholar 

  • McCarthy, B.P., Howell, K.C.: Leveraging quasi-periodic orbits for trajectory design in cislunar space. Astrodynamics 5(2), 139–165 (2021). https://doi.org/10.1007/s42064-020-0094-5

    Article  Google Scholar 

  • McCartin, B.J.: A matrix analytic approach to conjugate diameters of an ellipse. Appl. Math. Sci. 7(36), 1797–1810 (2013)

    MathSciNet  Google Scholar 

  • Meiss, J.D.: Differential Dynamical Systems, Chapter 9: Hamiltonian Dynamics, pp. 333–391 (2007)

  • Montenbruck, O., Kahle, R., D’Amico, S., Ardaens, J.-S.: Navigation and control of the TanDEM-X formation. J. Astronaut. Sci. 56(3), 341–357 (2008)

    Article  ADS  Google Scholar 

  • Noble, B., Daniel, J.W.: Applied Linear Algebra. Prentice-Hall, Englewood Cliffs (1969)

    MATH  Google Scholar 

  • Olikara, Z.P., Scheeres, D.J.: Numerical methods for computing quasi-periodic orbits and their stability in the restricted three-body problem. In: Advances in the Astronautical Sciences, pp. 911–930 (2012)

  • Roscoe, C.W., Vadali, S.R., Alfriend, K.T.: Third-body perturbation effects on satellite formations. J. Astronaut. Sci. 60(3), 408–433 (2013). https://doi.org/10.1007/s40295-015-0057-x

    Article  Google Scholar 

  • Schaub, H.: Relative orbit geometry through classical orbit element differences. J. Guid. Control Dyn. 27(5), 839–848 (2004). https://doi.org/10.2514/1.12595

    Article  ADS  Google Scholar 

  • Schaub, H., Junkins, J.: Analytical mechanics of space systems, Chapter 14: Spacecraft Formation Flying, pp. 709–792. AIAA Education Series. American Institute of Aeronautics and Astronautics, Incorporated (2014). https://doi.org/10.2514/4.102400

  • Scheeres, D.J., Hsiao, F.-Y., Vinh, N.X.: Stabilizing motion relative to an unstable orbit: applications to spacecraft formation flight. J. Guid. Control Dyn. 26(1), 62–73 (2003). https://doi.org/10.2514/2.5015

    Article  ADS  Google Scholar 

  • Schnizer, P., Fischer, E., Schnizer, B.: Cylindrical circular and elliptical, toroidal circular and elliptical multipoles fields, potentials and their measurement for accelerator magnets. arXiv:1410.8090 (2014)

  • Seeger, M., Hetzel, A., Dai, Z., Meissner, E., Lawrence, N.D.: Auto-differentiating linear algebra. arXiv:1710.08717 (2017)

  • Sengupta, P., Vadali, S.R.: Relative motion and the geometry of formations in Keplerian elliptic orbits with arbitrary eccentricity. J. Guid. Control Dyn. 30(4), 953–964 (2007). https://doi.org/10.2514/1.25941

    Article  ADS  Google Scholar 

  • Short, C.R., Blazevski, D., Howell, K.C., Haller, G.: Stretching in phase space and applications in general nonautonomous multi-body problems. Celest. Mech. Dyn. Astron. 122, 213–238 (2015). https://doi.org/10.1007/s10569-015-9617-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Simó, C., Gómez, G., Llibre, J., Martínez, R., Rodríguez, J.: On the optimal station keeping control of halo orbits. Acta Astronaut. 15(6–7), 391–397 (1987)

    Article  ADS  Google Scholar 

  • Spergel, D., Gehrels, N., Baltay, C., Bennett, D., Breckinridge, J., Donahue, M., Dressler, A., Gaudi, B., Greene, T., Guyon, O., et al.: Wide-Field Infrarred Survey Telescope-Astrophysics Focused Telescope Assets. arXiv:1503.03757 (2015)

  • Sullivan, J., Grimberg, S., D’Amico, S.: Comprehensive survey and assessment of spacecraft relative motion dynamics models. J. Guid. Control Dyn. 40(8), 1837–1859 (2017). https://doi.org/10.2514/1.G002309

    Article  ADS  Google Scholar 

  • Szebehely, V.: Theory of Orbits. Academic Press, London (1967)

    MATH  Google Scholar 

  • Townsend, J.: Differentiating the singular value decomposition. Technical Report (2016). Accessed 29 April 2021

  • Tschauner, J., Hempel, P.: Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel. Acta Astronaut. 11(2), 104–109 (1965)

    MATH  Google Scholar 

  • Vallado, D.A., McClain, W.D.: Fundamentals of Astrodynamics and Applications, 4th edn. Space Technology Library, Microcosm Press (2013)

  • Wiesel, W., Shelton, W.: Modal control of an unstable periodic orbit. J. Astronaut. Sci. (1983)

Download references

Acknowledgements

This work was supported by a Graduate Assistance in Areas of National Need fellowship from the Smead Department of Aerospace Engineering Sciences at the University of Colorado Boulder. An earlier version of this work was presented as AAS 20-620 at the AAS/AIAA Astrodynamics Specialist Virtual Conference in August 2020. The authors also wish to thank each of the reviewers for their feedback.

Funding

This work was supported by a Graduate Assistance in Areas of National Need fellowship from the Smead Department of Aerospace Engineering Sciences at the University of Colorado Boulder.

Author information

Authors and Affiliations

  1. Colorado Center for Astrodynamics Research, Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, 429 UCB, 3775 Discovery Drive, Boulder, CO, 80303, USA

    Ian Elliott & Natasha Bosanac

Authors
  1. Ian Elliott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Natasha Bosanac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ian Elliott.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Code availability

No additional codes are provided.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elliott, I., Bosanac, N. Describing relative motion near periodic orbits via local toroidal coordinates. Celest Mech Dyn Astr 134, 19 (2022). https://doi.org/10.1007/s10569-022-10074-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10569-022-10074-8

Keywords

Search

Navigation