Formation flight design for a LISA-like gravitational wave observatory via Cascade optimization

  • Chihang Yang
  • Hao ZhangEmail author
Research Article


Laser Interferometer Space Antenna (LISA) is a project to detect and measure gravitational waves. The project has three spacecraft flying in a formation of near equilateral triangle in a heliocentric orbit trailing Earth. Many sources of perturbations cause the configuration to deviate from the nominal. This paper studies the formation design problem for a LISA-like mission by considering ephemeris-based dynamics. This type of mission is well-known for addressing several strict mission requirements under the realistic dynamics. The problem is formulated as optimizing multiple mission performance indices. It is observed that some indices are correlated with each other, whereas some indices have different sensitivities with respect to the semi-major axis. Therefore, the problem is transformed into a two-step cascade single-objective optimization, in which the optimal solution of the first optimization problem is fed to the second optimization as initial value. In addition, the major perturbing celestial bodies are picked up to make a simplified but accurate enough dynamics to speed up the optimization. Numerical examples verify the analysis and show the effectiveness of the optimization procedure. The influences of mission lifetime and spatial scales on the optimal solutions are also presented.


modified equinoctial orbit elements LISA spacecraft formation flight gravitational wave observatory Monte-Carlo multi-objective optimization 



The work was supported by the Hundred Talents Program of the Chinese Academy of Sciences and Strategic Priority Program A (Grant No. XDA15014902).


  1. [1]
    Rowan, S., Hough, J. Gravitational wave detection by interferometry (ground and space). Living Reviews in Relativity, 2000, 3(1): 3.zbMATHGoogle Scholar
  2. [2]
    Danzmann, K. LISA laser interferometer space antenna—a proposal in response to the ESA call for L3 mission concepts. Technical report, ESA, 2017.Google Scholar
  3. [3]
    Men, J. R., Ni, W. T., Wang, G. Design of ASTROD-GW orbit. Chinese Astronomy and Astrophysics, 2010, 34(4): 434–446.Google Scholar
  4. [4]
    Kawamura, S., Nakamura, T., Ando, M., Seto, N., Tsubono, K., Numata, K., Takahashi, R., Nagano, S., Ishikawa, T., Musha, M. et al. The Japanese space gravitational wave antenna-DECIGO. Classical and Quantum Gravity, 2006, 23(8): 125–131.MathSciNetGoogle Scholar
  5. [5]
    Gong, X. F., Lau, Y. K., Xu, S. N., Amaro-Seoane, P., Bai, S., Bian, X., Cao, Z. J., Chen, G. R., Chen, X., Ding, Y. W. et al. Descope of the ALIA mission. Journal of Physics: Conference Series, 2015, 610(1): 012011.Google Scholar
  6. [6]
    Luo, J., Chen, L. S., Duan, H. Z., Gong, Y. G., Hu, S., Ji, J. H., Liu, Q., Mei, J. W., Milyukov, V., Sazhin, M. et al. TianQin: a space-borne gravitational wave detector. Classical and Quantum Gravity, 2016, 33(3): 035010.Google Scholar
  7. [7]
    Diaz-Aguiló, M., Mateos, I., Ramos-Castro, J., Lobo, A., García-Berro, E. Design of the magnetic diagnostics unit onboard LISA pathfinder. Aerospace Science and Technology, 2013, 26(1): 53–59.Google Scholar
  8. [8]
    Trenkel, C., Kemble, S., Bevis, N., Magueijo, J. Testing modified newtonian dynamics with LISA pathfinder. Advances in Space Research, 2012, 50(11): 1570–1580.Google Scholar
  9. [9]
    Ziegler, T., Bergner, P., Hechenblaikner, G., Brandt, N., Fichter, W. Modeling and performance of contact-free discharge systems for space inertial sensors. IEEE Transactions on Aerospace and Electronic Systems, 2014, 50(2): 1493–1510.Google Scholar
  10. [10]
    Ni, W. T. Gravitational wave detection in space. International Journal of Modern Physics D, 2016, 25(14): 1630001.Google Scholar
  11. [11]
    ESA. The ESA-L3 gravitational wave mission — gravitational observatory advisory team — Final Report. Technical report, ESA, 2016.Google Scholar
  12. [12]
    Jennrich, O., Binétruy, P., Colpi, M., Danzmann, K., Jetzer, P., Lobo, A., Nelemans, G., Schutz, B. F., Stebbins, R., Sumner, T. et al. NGO: Revealing a hidden universe: opening a new chapter of discovery. Technical report ESA/SRE(2011)19, ESA, 2011.Google Scholar
  13. [13]
    Dhurandhar, S., Nayak, K. R., Koshti, S., Vinet, J. Y. Fundamentals of the LISA stable flight formation. Classical and Quantum Gravity, 2005, 22(3): 481–487.zbMATHGoogle Scholar
  14. [14]
    De Marchi, F., Pucacco, G., Bassan, M. Optimizing the Earth-LISA ‘rendezvous’. Classical and Quantum Gravity, 2012, 29(3): 035009.zbMATHGoogle Scholar
  15. [15]
    Alfriend, K. T., Vadali, S. R., Gurfil, P., How, J., Breger, L. S. Spacecraft formation flying: dynamics, control and navigation. Butterworth-Heinemann, 2009.Google Scholar
  16. [16]
    Nayak, K. R., Koshti, S., Dhurandhar, S. V., Vinet, J. Y. On the minimum flexing of LISA’s arms. Classical and Quantum Gravity, 2006, 23(5): 1763–1778.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Li, G. Y., Yi, Z. H., Heinzel, G., Rüdigger, A., Jennrich, O., Wang, L., Xia, Y., Zeng, F., Zhao, H. B. Methods for orbit optimization for the LISA gravitational wave observatory. International Journal of Modern Physics D, 2008, 17(7): 1021–1042.zbMATHGoogle Scholar
  18. [18]
    Yi, Z. H., Li, G. Y., Heinzel, G., Rüdigger, A., Luo, Y. J., Xia, Y., Zhao, H. B. The Co-orbital restricted three-body problem and its application. Science China Physics, Mechanics and Astronomy, 2010, 53(1): 171–178.Google Scholar
  19. [19]
    Pucacco, G., Bassan, M., Visco, M. Autonomous perturbations of LISA orbits. Classical and Quantum Gravity, 2010, 27(23): 235001.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Cerdonio, M., De Marchi, F., De Pietri, R., Jetzer, P., Marzari, F., Mazzolo, G., Ortolan, A., Sereno, M. Modulation of LISA free-fall orbits due to the Earth-Moon system. Classical and Quantum Gravity, 2010, 27(16): 165007.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Hughes, S. P., Bauer, F. H. Preliminary optimal orbit design for the laser interferometer space antenna (LISA). Technical report, NASA, 2002.Google Scholar
  22. [22]
    Halloin, H. Optimizing orbits for (e)LISA. Journal of Physics: Conference Series, 2017, 840(1): 012048.Google Scholar
  23. [23]
    Walker, M. J. H., Ireland, B., Owens, J. A set modified equinoctial orbit elements. Celestial Mechanics and Dynamical Astronomy, 1985, 36(4): 409–419.zbMATHGoogle Scholar
  24. [24]
    Gim, D. W., Alfriend, K. T. Satellite relative motion using differential equinoctial elements. Celestial Mechanics and Dynamical Astronomy, 2005, 92(4): 295–336.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Povoleri, A., Kemble, S. LISA orbits. AIP Conference Proceedings, 2006, 873: 702–706.Google Scholar
  26. [26]
    Giulicchi, L., Wu, S. F., Fenal, T. Attitude and orbit control systems for the LISA pathfinder mission. Aerospace Science and Technology, 2013, 24(1): 283–294.Google Scholar
  27. [27]
    Xia, Y., Li, G. Y., Heinzel, G., Rüdigger, A., Luo, Y. J. Orbit design for the laser interferometer space antenna (LISA). Science China Physics, Mechanics and Astronomy, 2010, 53(1): 179–186.Google Scholar
  28. [28]
    Gurfil, P. Relative motion between elliptic orbits: generalized boundedness conditions and optimal formationkeeping. Journal of Guidance, Control, and Dynamics, 2005, 28(4): 761–767.MathSciNetGoogle Scholar
  29. [29]
    Wang, G., Ni, W. T. Numerical simulation of time delay interferometry for new LISA, TAIJI and other LISA-like missions. arXiv:1707.09127, 2017.Google Scholar
  30. [30]
    Bik, J. J. C. M., Visser, P. N. A. M., Jennrich, O. LISA satellite formation control. Advances in Space Research, 2007, 40(1): 25–34.Google Scholar
  31. [31]
    Sweetser, T. H. An end-to-end trajectory description of the LISA mission. Classical and Quantum Gravity, 2005, 22(10): 429–435.Google Scholar
  32. [32]
    Merkowitz, S. M., Ahmad, A., Hyde, T. T., Sweetser, T., Ziemer, J., Conkey, S., Kelly III, W., Shirgur, B. LISA propulsion module separation study. Classical and Quantum Gravity, 2005, 22(10): 413–419.Google Scholar

Copyright information

© Tsinghua University Press 2019

Authors and Affiliations

  1. 1.Technology and Engineering Center for Space UtilizationChinese Academy of SciencesBeijingChina
  2. 2.Key Laboratory of Space Utilization, Technology and Engineering Center for Space UtilizationChinese Academy of SciencesBeijingChina

Personalised recommendations