Skip to main content
SpringerLink
Log in

Curvilinear coordinate transformations for relative motion

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

An Erratum to this article was published on 17 March 2016

Abstract

Relative motion improvements have traditionally focused on inserting additional force models into existing formulations to achieve greater fidelity, or complex expansions to admit eccentric orbits for propagation. A simpler approach may be numerically integrating the two satellite positions and then converting to a modified equidistant cylindrical frame for comparison in a Hill’s-like frame. Recent works have introduced some approaches for this transformation within the Hill’s construct, and examined the accuracy of the transformation. Still others have introduced transformations as they apply to covariance operations. Each of these has some orbital or force model limitations and defines an approximate circular reference dimension. We develop a precise transformation between the Cartesian and curvilinear frame along the actual satellite orbit and test the results for various orbital classes. The transformation has wide applicability.

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

Similar content being viewed by others

Notes

  1. We use Hill’s (1878) and Clohessy–Wiltshire (CW, Clohessy and Wiltshire 1960) interchangeably.

  2. We use the term “sigma points” to indicate the depiction of an ellipsoid by its 6 principal points. These points fully describe the ellipsoid and they lie along the eigen-axes of the ellipsoid. There are two methods we use. One finds the 6 points to describe the ellipsoid (eigenvalue decomposition), and one to find the ellipsoid given just the 6 points (minimum volume enclosing ellipsoid algorithm).

  3. Surface points are used in the inverse transformation because using the sigma points would yield the same ellipsoid as in Fig. 8. The number of surface points taken depends on the desired grid and fineness of the resulting ellipsoid.

References

  • Alfriend, K.T., Yan, H.: An evaluation and comparison of relative motion theories. AIAA J. Guid. Control Dyn. 26(2), 254–263 (2005)

    Article  ADS  Google Scholar 

  • Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–674 (1960)

    Article  MATH  Google Scholar 

  • Fasano, G., D’Errico, M.: Modeling orbital relative motion to enable formation design from application requirements. Celest. Mech. Dyn. Astron. 105, 113–139 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  • Gomez, G., Marcote, M.: High-order analytical solutions of Hill’s equations. Celest. Mech. Dyn. Astron. 94, 197–211 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hill, K., Alfriend, K.T., Sabol, C.: Covariance-based uncorrelated track association. In: Paper AIAA 2008–7211 Presented at the AIAA/AAS Astrodynamics Specialist Conference. Honolulu, HI. August 18–21 (2008)

  • Hill, K. et al.: Covariance-based network tasking of optical sensors. In: Paper AAS 10–150 Presented at the AAS/AIAA Space Flight Mechanics Conference, February 14–17. San Diego, CA. (Also 8th USR Workshop, Maui, April 8–13, 2010, Paper S5.2) (2010)

  • Hill. K., Sabol, C., Alfriend, K.T.: Comparison of covariance-based track association approaches with simulated radar data. In: Paper AAS 10–318 Presented at the Kyle T. Alfriend Astrodynamics Symposium. Monterey, CA. May 17–19 (2010)

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

    Article  MathSciNet  MATH  Google Scholar 

  • Junkins, J.L., Akella, M.R., Alfriend, K.T.: Non-Gaussian error propagation in orbital mechanics. J. Astronaut. Sci. 44(4), 541–563 (1996)

    Google Scholar 

  • Martinusi, V., Gurfil, P.: Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations. Celest. Mech. Dyn. Astron. 111, 387–414 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  • Moiseev, I.: Personal Discussions. Code at http://elliptic.googlecode.com, Accessed June 2013 (2013)

  • Park, R.S., Scheeres, D.J.: Nonlinear mapping of Gaussian statistics: theory and applications to spacecraft trajectory design. J. Guid. Control Dyn. 29(6), 1367–1375 (2006)

    Article  ADS  Google Scholar 

  • Sabol, C. et al.: Linearized orbit covariance generation and propagation analysis via simple Monte Carlo simulations. In: Paper AAS 10–134 Presented at the AAS/AIAA Space Flight Mechanics Conference, February 14–17. San Diego, CA. (Also 8th USR Workshop, Maui, April 8–13, 2010, Paper S4.1) (2010)

  • Schweighart, S.A., Sedwick, R.J.: High fidelity linearized J2 model for satellite formation flight. J. Guid. Control Dyn. 25(6), 1073–1080 (2002)

    Article  ADS  Google Scholar 

  • Tschauner, J., Hempel, P.: Rendezvous with a target in an elliptical orbit. Astronaut. Acta 11(2), 104–109 (1965)

    MATH  Google Scholar 

  • Vallado, D.A.: Fundamentals of Astrodynamics and Applications, 1st edn. McGraw Hill, New York (1997)

  • Vallado, D.A.: Fundamentals of Astrodynamics and Applications, 4th edn. Microcosm, Hawthorne, CA (2013)

    MATH  Google Scholar 

  • Weisstein, E.W. (2011). Cylindrical equidistant projection. From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/CylindricalEquidistantProjection.html

Download references

Author information

Authors and Affiliations

  1. Center for Space Standards and Innovation, Analytical Graphics Inc., 7150 Campus Dr., Suite 260, Colorado Springs, CO , 80920-6522, USA

    David A. Vallado & Salvatore Alfano

Authors
  1. David A. Vallado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salvatore Alfano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David A. Vallado.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vallado, D.A., Alfano, S. Curvilinear coordinate transformations for relative motion. Celest Mech Dyn Astr 118, 253–271 (2014). https://doi.org/10.1007/s10569-014-9531-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-014-9531-1

Keywords

Search

Navigation