Celestial Mechanics and Dynamical Astronomy

, Volume 121, Issue 3, pp 275–300 | Cite as

Extended analytical formulas for the perturbed Keplerian motion under a constant control acceleration

  • Federico Zuiani
  • Massimiliano VasileEmail author
Original Article


This paper presents a set of analytical formulae for the perturbed Keplerian motion of a spacecraft under the effect of a constant control acceleration. The proposed set of formulae can treat control accelerations that are fixed in either a rotating or inertial reference frame. Moreover, the contribution of the \(J_{2}\) zonal harmonic is included in the analytical formulae. It will be shown that the proposed analytical theory allows for the fast computation of long, multi-revolution spirals while maintaining good accuracy. The combined effect of different perturbations and of the shadow regions due to solar eclipse is also included. Furthermore, a simplified control parameterisation is introduced to optimise thrusting patterns with two thrust arcs and two cost arcs per revolution. This simple parameterisation is shown to ensure enough flexibility to describe complex low thrust spirals. The accuracy and speed of the proposed analytical formulae are compared against a full numerical integration with different integration schemes. An averaging technique is then proposed as an application of the analytical formulae. Finally, the paper presents an example of design of an optimal low-thrust spiral to transfer a spacecraft from an elliptical to a circular orbit around the Earth.


Low-thrust trajectories Analytical solutions First order expansions  Orbit averaging Orbit circularisation 



The authors would like to thank Dr. Richard Epenoy from CNES for his helpful suggestions and for the results of the test case with MIPELEC.

Supplementary material

10569_2014_9600_MOESM1_ESM.pdf (52 kb)
Supplementary material 1 (pdf 51 KB)


  1. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, New York (1987)zbMATHGoogle Scholar
  2. Beletsky, V.V.: Essays on the Motion of Celestial Bodies. Birkhäuser Verlag, Basel (1999)Google Scholar
  3. Bombardelli, C., Baù, G., Peláez, J.: Asymptotic solution for the two-body problem with constant tangential thrust acceleration. Celest. Mech. Dyn. Astron. 110(3), 239–256 (2011)Google Scholar
  4. Carlson, B.C.: Computing elliptic integrals by duplication. Numer. Math. 33(1), 1–16 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Casalino, L., Colasurdo, G.: Improved Edelbaum’s approach to optimize low Earth/geostationary orbits low-thrust transfers. J. Guid. Control Dyn. 30(5), 1504–1510 (2007)ADSCrossRefGoogle Scholar
  6. Colombo, C., Vasile, M., Radice, G.: Semi-analytical solution for the optimal low-thrust deflection of near-Earth objects. J. Guid. Control Dyn. 32(3), 796–809 (2009)ADSCrossRefGoogle Scholar
  7. Colombo, C., McInnes, C.: Orbital dynamics of “Smart-Dust” devices with solar radiation pressure and drag. J. Guid. Control Dyn. 34(6), 1613–1631 (2011)ADSCrossRefGoogle Scholar
  8. Escobal, P.: Methods of Orbit Determination. Wiley, New York (1965)Google Scholar
  9. Evtushenko, I.G.: Influence of tangential acceleration on the motion of a satellite. J. Appl. Math. Mech. 30(3), 710–716 (1967)Google Scholar
  10. Ferrier, C., Epenoy, R.: Optimal control for engines with electro-ionic propulsion under constraint of eclipse. Acta Astronaut. 48(4), 181–192 (2001)ADSCrossRefGoogle Scholar
  11. Gao, Y., Li, X.: Optimization of low-thrust many-revolution transfers and Lyapunov-based guidance. Acta Astronaut. 66(1), 117–129 (2010)ADSCrossRefGoogle Scholar
  12. Geffroy, S., Epenoy, R.: Optimal low-thrust transfers with constraints, generalization of averaging techniques. Acta Astronaut. 41(3), 133–149 (1997)ADSCrossRefGoogle Scholar
  13. Holmes, M.H.: Introduction to the Foundations of Applied Mathematics. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  14. Kechichian, J.A.: Reformulation of Edelbaum’s low-thrust transfer problem using optimal control theory. J. Guid. Control Dyn. 20(5), 988–994 (1997a)Google Scholar
  15. Kechichian, J.A.: The treatment of the Earth oblateness effect in trajectory optimization in equinoctial coordinates. Acta Astronaut. 40, 69–82 (1997b)ADSCrossRefGoogle Scholar
  16. Kechichian, J.A.: Low-thrust eccentricity-constrained orbit raising. J. Spacecr. Rockets 35(3), 327–335 (1998a)ADSCrossRefGoogle Scholar
  17. Kechichian, J.A.: Orbit raising with low-thrust tangential acceleration in presence of Earth shadow. J. Spacecr. Rockets 35(4), 516–525 (1998b)ADSCrossRefGoogle Scholar
  18. Kechichian, J.A.: Low-thrust inclination control in presence of Earth shadow. J. Spacecr. Rockets 35(4), 526–532 (1998c)ADSCrossRefGoogle Scholar
  19. Kechichian, J.A.: The streamlined and complete set of the nonsingular J2-perturbed dynamic and adjoint equations for trajectory optimization in terms of eccentric longitude. J. Astronaut. Sci. 55(3), 325–348 (2007)ADSCrossRefMathSciNetGoogle Scholar
  20. Kluever, C.A., Oleson, S.R.: Direct approach for computing near-optimal low-thrust Earth-orbit transfers. J. Spacecr. Rockets 35(4), 509–515 (1998)ADSCrossRefGoogle Scholar
  21. Lantoine, G., Russell, R.P.: The Stark model: an exact, closed-form approach to low-thrust trajectory optimization. In: 21st International Symposium on Space Flight Dynamics (2009)Google Scholar
  22. Lantoine, G., Russell, R.P.: Complete closed-form solutions of the Stark problem. Celest. Mech. Dyn. Astron. 109(4), 333–366 (2011)Google Scholar
  23. Prince, P.J., Dormand, J.R.: High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  24. Russell, C.T., Capaccioni, F., Coradini, A., De Sanctis, M.C., Feldman, W.C., Jaumann, R., et al.: Dawn mission to Vesta and Ceres. EarthMoon Planets 101(1–2), 65–91 (2007)Google Scholar
  25. Sanders, J.J.A., Verhulst, F., Murdock, J.A.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (2007)zbMATHGoogle Scholar
  26. Schoenmaekers, J., Horas, D., Pulido, J.A.: SMART-1: With solar electric propulsion to the Moon. In: International Symposium on Space Flight Dynamics, Pasadena, CA, USA (2001)Google Scholar
  27. Smith, J.C., Parcher, D.W., Whiffen, G.J.: Spiraling away from Vesta: design of the transfer from the low to high altitude Dawn mapping orbits. In: 23rd AAS/AIAA space flight mechanics conference, Kauai, Hawaii, US (2013)Google Scholar
  28. Tarzi, Z., Speyer, J., Wirz, R.: Fuel optimum low-thrust elliptic transfer using numerical averaging. Acta Astronaut. 86, 95–118 (2013)ADSCrossRefGoogle Scholar
  29. Vallado, D.A.: Fundamentals of Astrodynamics and Applications, 3rd edn, Space Technology Library. Springer, New York (2007)Google Scholar
  30. Zuiani, F., Vasile, M.: Preliminary design of Debris removal missions by means of simplified models for low-thrust, many-revolution transfers. In: International Journal of Aerospace Engineering, Hindawi (2012)Google Scholar
  31. Zuiani, F., Vasile, M., Palmas, A., Avanzini, G.: Direct transcription of low-thrust trajectories with finite trajectory elements. In: 61th International Astronautical Congress of the International Astronautical Federation, Prague, Czech Republic, IAC-10-C1.7.5 (2010)Google Scholar
  32. Zuiani, F., Vasile, M., Palmas, A., Avanzini, G.: Direct transcription of low-thrust trajectories with finite trajectory elements. Acta Astronaut. 72, 108–120 (2012a)ADSCrossRefGoogle Scholar
  33. Zuiani, F., Vasile, M., Gibbings, A.: Evidence-based robust design of deflection actions for near Earth objects. Celest. Mech. Dyn. Astron. 114(1–2), 107–136 (2012b)ADSCrossRefMathSciNetGoogle Scholar
  34. Zuiani, F., Kawakatsu, Y., Vasile, M.: Multi-objective optimisation of many-revolution, low-thrust orbit raising for destiny mission. In: 23rd AAS/AIAA space flight mechanics conference, Kauai, Hawaii, US (2013)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.University of GlasgowGlasgowUK
  2. 2.University of StrathclydeGlasgowUK

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