Skip to main content

Reachable domain for spacecraft with ellipsoidal Delta-V distribution

Astrodynamics volume 2, pages 265–288 (2018)Cite this article

Abstract

Conventional reachable domain (RD) problem with an admissible velocity increment, Δv, in an isotropic distribution, was extended to the general case with Δv in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Vinh, N., Gilbert, E., Howe, R., Sheu, D., Lu, P. Reachable domain for interception at hyperbolic speeds. Acta Astronautica, 1995, 35 (1): 1–8.

    Article  Google Scholar 

  2. Xue, D., Li, J., Baoyin, H. Study on reachable domain for satellite trajectory with coplanar impulse applied, Journal of Astronautics, 2009, 30(1): 88–92.

    Google Scholar 

  3. Xue, D., Li, J., Jiang, F. Reachable domain of a satellite with a coplanar impulse applied, Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(2): 337–342.

    Google Scholar 

  4. Xue, D., Li, J., Baoyin, H., Jiang, F. Reachable domain for spacecraft with a single impulse. Journal of Guidance, Control, and Dynamics, 2010, 33(3): 934–942.

    Article  Google Scholar 

  5. Li, X., He, X., Zhong, Q. Song, M. Reachable domain for satellite with two kinds of thrust. Acta Astronautica, 2011, 68(11-12): 1860–1864.

    Article  Google Scholar 

  6. Wen, C., Zhao, Y., Shi, P., Hao, Z., Orbital accessibility problem for spacecraft with a single impulse. Journal of Guidance, Control, and Dynamics, 2014, 37: 1260–1271.

    Article  Google Scholar 

  7. Wen, C., Zhao, Y., Shi, P. Precise determination of reachable domain for spacecraft with single impulse. Journal of Guidance, Control, and Dynamics, 2014, 37(4): 1767–1779.

    Article  Google Scholar 

  8. Zhang, G., Cao, X., Ma, G. Reachable domain of spacecraft with a single tangent impulse considering trajectory safety. Acta Astronautica, 2013, 91: 228–236.

    Article  Google Scholar 

  9. Wen, C., Gurfil, P. Modeling early medium-term evolution of debris clouds using the reachable domain method. Journal of Guidance, Control, and Dynamics, 2016, 39(12): 2649–2660.

    Article  Google Scholar 

  10. Breakwell, J. V., Heine, W. Minimum-impulse, timefree transfer between neighboring noncoplanar elliptical orbits with major axes perpendicular to the line of nodes. AIAA Journal, 1969, 7(7):1236–1241.

    Article  MATH  Google Scholar 

  11. Marec, J. P. Optimal space trajectories, Elsevier, 2012, 1.

    Google Scholar 

  12. Holzinger, M., Scheeres, D. On-Orbit Range Set Applications. Advanced Maui Optical and Space Surveillance Technologies Conference, 2011.

    Google Scholar 

  13. Holzinger, M. J., Scheeres, D. Reachability results for nonlinear systems with ellipsoidal initial sets. IEEE transactions on aerospace and electronic systems, 2012, 48(2): 1583–1600.

    Article  Google Scholar 

  14. Holzinger, M., Scheeres, D. J., Erwin, R. S. On-Orbit operational range computation using gaussars variational equations with J2 perturbations. Journal of Guidance, Control, and Dynamics, 2014, 37(2): 608–622.

    Article  Google Scholar 

  15. Holzinger, M., Scheeres, D., Hauser, J. Optimal reachability sets using generalized independent parameters. Proceedings of the 2011 American Control Conference, 2012.

    Google Scholar 

  16. Ribeiro, M. I. Gaussian probability density functions: properties and error characterization. Proceedings of the Institute for Systems and Robotics, 2004.

    Google Scholar 

  17. Mahalanobis, P. C. On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, 1936.

    Google Scholar 

  18. Battin, R. H. An introduction to the mathematics and methods of astrodynamics. AIAA, 1999.

    Google Scholar 

  19. Apostol, T. M., Mathematical Analysis. Addison-Wesley Publishing Company, 1974.

    Google Scholar 

  20. Weisstein, E. First derivative test. The MathWorld-A Wolfram Web Resource, 2016, http://mathworld.wolfram.com/FirstDerivativeTest.Html

    Google Scholar 

  21. Driscoll, T. A., Hale, N., Trefethen, L. N. Chebfun guide. Pafnuty Publications, 2014.

    Google Scholar 

  22. Vallado, D. A., McClain, W. D. Fundamentals of astrodynamics and applications. Springer, 2001.

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11702293).

Author information

Authors and Affiliations

  1. Chinese Academy of Sciences, Beijing, 100094, China

    Changxuan Wen, Chao Peng & Yang Gao

Authors
  1. Changxuan Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changxuan Wen.

Additional information

Changxuan Wen received his B.S. and Ph.D. degrees from Beihang University, China, in 2009 and 2016, respectively, and was a visiting scholar of Technion- Israel Institute of Technology, Israel, in 2014 and 2015. Since June 2015, he has been an associate research fellow at Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include flight dynamics, satellite conjunction analysis, geostationary servicing, reachable domain, and guidance and navigation. Email: wenchangxuan@gmail.com

Yang Gao received his B.S. and M.S. degrees from Beihang University and Chinese Academy of Sciences in 1997 and 2000 respectively. He received his Ph.D. degree, in 2003 from the University of Missouri, Columbia, USA. From 2004 to 2005, he was a postdoctoral research fellow from the University of Missouri. Since June 2005, he has been an associate research fellow at Academy of Opto-Electronic, Chinese Academy of Sciences. Currently, he is a research professor at Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include orbit determination, spacecraft dynamics and control, guidance and navigation, trajectory optimization, and space mission design. Email: gaoyang@csu.ac.cn

Chao Peng received his B.S. degree from the Taiyuan University of Technology in 2008 and his M.S. degree from the Chinese Academy of Sciences in 2012. Since June 2012, he has been a junior engineer at the Technology and Engineering Center of Space Utilization, Chinese Academy of Sciences. His academic interests include spacecraft dynamics and control, trajectory optimization, and space mission design. Email: pc309@csu.ac.cn

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wen, C., Peng, C. & Gao, Y. Reachable domain for spacecraft with ellipsoidal Delta-V distribution. Astrodyn 2, 265–288 (2018). https://doi.org/10.1007/s42064-018-0025-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42064-018-0025-x

Keywords

  • reachable domain
  • maneuverability
  • orbit uncertainty