Skip to main content

Impact of Using Analytic Derivatives In Optimization For N-Impulse Orbit Transfer Problems


Several formulations are possible for the optimization of N-impulse two-body orbit transfers. One formulation that assumes the first N − 1 impulses are design variables, and implements Lambert’s algorithm in the final leg is considered here. This paper presents a derivation for the analytic expressions of the gradients needed to optimize a transfer using this formulation. The derivations of the analytic gradients, verification tests using complex-step differentiation, as well as numerical case studies for three-impulse orbit transfers are presented. The numerical case studies highlight a significant reduction in the computational cost, measured in terms of the number of objective function evaluations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


Δ v∥:

sum of magnitudes of instantaneous velocity changes (cost function), DU/TU

r :

position vector, DU

v :

velocity vector, DU/TU

Δ t :

time of flight, TU

t :

time on orbit, TU

𝜃 :

true anomaly, rad

Δ 𝜃 :

difference between two true anomalies, rad

E :

eccentric anomaly, rad

H :

hyperbolic eccentric anomaly, rad

n :

mean motion, rad/s

e :


p :

orbit parameter, DU

μ :

gravitational parameter, DU3/TU2

h :

specific angular momentum, DU2/TU

a :

semi-major axis, DU

\(\hat {p}, \hat {q} , \hat {w}\) :

perifocal frame

\(f, g, \bar {f}\) :

Lagrange Coefficients

t pi :

time from perigee until point i, TU

c :

cord, DU

s :

half of the perimeter, DU

x :

dummy variable

α,β :

Lagrange Parameters

N :

total number of impulses

H :

hyperbolic orbit

i,f :

initial and final orbits


point on orbit


before impulse


after impulse


  1. Abdelkhalik, O., Mortari, D.: N-impulse orbit transfer using genetic algorithms. J. Spacecr. Rocket. 44(2), 456–460 (2007).

    Article  Google Scholar 

  2. Arora, N., Russell, R.P.: A fast and robust multiple revolution lambert algorithm using a cosine transformation. In: AAS/AIAA Astrodynamics Specialist Conference, AAS 13-728 (2013)

    Google Scholar 

  3. Arora, N., Russell, R.P., Strange, N., Ottesen, D.: Partial derivatives of the solution to the lambert boundary value problem. J. Guid. Contr. Dyn. 38(9), 1563–1572 (2015).

    Article  Google Scholar 

  4. Battin, R.H.: An introduction to the mathematics and methods of astrodynamics, rev. edn. American Institute of Aeronautics and Astronautics (1999).

  5. Colasurdo, G., Pastrone, D.: Indirect optimization method for impulsive transfers. In: Astrodynamics Conference, AIAA 1994-3762 (1994).

  6. Conway, B.: Spacecraft trajectory optimization. Cambridge University Press (2009).

  7. Ellison, D.H., Conway, B.A., Englander, J.A., Ozimek, M.T.: Analytic gradient computation for bounded-impulse trajectory models using two-sided shooting. J. Guid. Contr. Dyn. 41(7), 1449–1462 (2018).

    Article  Google Scholar 

  8. Ellithy, A., Abdelkhalik, O., Englander, J.: Impact of analytic derivatives on optimization of n-impulse orbit transfers. In: AAS/AIAA Astrodynamics Specialist Conference, AAS 20-491 (2020)

  9. Gong, M., Zhou, D., Shao, C., Fang, Y.: Optimal multiple-impulse time-fixed rendezvous using evolutionary algorithms. J. Spacecr. Rocket., 1–7 (2021).

  10. Herman, A.L., Conway, B.A.: Direct optimization using collocation based on high-order gauss-lobatto quadrature rules. J. Guid. Contr. Dyn. 19(3), 592–599 (1996).

    Article  Google Scholar 

  11. Lantoine, G., Russell, R.P., Dargent, T.: Using multicomplex variables for automatic computation of high-order derivatives. ACM Trans. Math. Softw. 38(3), 1–21 (2012).

    MathSciNet  Article  Google Scholar 

  12. Lawden, D.F.: Optimal trajectories for space navigation, vol. 3. Butterworths (1963)

  13. Luo, Y. Z., Zhang, J., Li, H.Y., Tang, G.J.: Interactive optimization approach for optimal impulsive rendezvous using primer vector and evolutionary algorithms. Acta Astronaut. 67(3-4), 396–405 (2010).

    Article  Google Scholar 

  14. Martins, J., Kroo, I., Alonso, J.: An automated method for sensitivity analysis using complex variables. In: 38th Aerospace Sciences Meeting and Exhibit, AIAA 2000-689 (2000).

  15. Martins, J., Sturdza, P., Alonso, J.: The complex-step derivative approximation. ACM Trans. Math. Softw. 29(3), 245–262 (2003)

    MathSciNet  Article  Google Scholar 

  16. Pellegrini, E., Russell, R.P.: On the computation and accuracy of trajectory state transition matrices. J. Guid. Contr. Dyn. 39(11), 2485–2499 (2016).

    Article  Google Scholar 

  17. Pontani, M., Ghosh, P., Conway, B.A.: Particle swarm optimization of multiple-burn rendezvous trajectories. J. Guid. Contr. Dyn. 35(4), 1192–1207 (2012).

    Article  Google Scholar 

  18. Prussing, J.: A class of optimal two-impulse rendezvous using multiple-revolution Lambert solutions. J. Astronaut. Sci. 48(2-3), 131–148 (2000).

    Article  Google Scholar 

  19. Prussing, J., Chiu, J.H.: Optimal multiple-impulse time-fixed rendezvous between circular orbits. In: Astrodynamics Conference, AIAA 1984-2036 (1984).

  20. Shen, H.X., Casalino, L., Luo, Y.Z.: Global search capabilities of indirect methods for impulsive transfers. J. Astronaut. Sci. 62(3), 212–232 (2015).

    Article  Google Scholar 

  21. Shirazi, A., Ceberio, J., Lozano, J.A.: Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions. Prog. Aerospace Sci. 102, 76–98 (2018).

    Article  Google Scholar 

  22. Vallado, D.A.: Fundamentals of Astrodynamics and Applications, 3rd edn. Space Technology Libary. Microcosm Press and Springer (2007)

  23. Zhang, G., Zhou, D., Mortari, D., Akella, M.R.: Covariance analysis of Lambert’s problem via Lagrange’s transfer-time formulation. Aerosp. Sci. Technol. 77, 765–773 (2018)

    Article  Google Scholar 

  24. Zhu, Y., Wang, H., Zhang, J.: Spacecraft multiple-impulse trajectory optimization using differential evolution algorithm with combined mutation strategies and boundary-handling schemes. Math. Probl. Eng. 2015, 1–13 (2015).

    Google Scholar 

Download references


Authors would like to thank Dr. Noble Hatten for his feedback and suggestions


This paper is based upon work supported by NASA, Award Number 80NSSC19K1642

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ahmed Ellithy.

Ethics declarations

Conflict of Interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

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

This paper is based on a conference paper AAS 20-491 presented at the the AAS/AIAA Astrodynamics Specialist Conference, August 2020 [8].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ellithy, A., Abdelkhalik, O. & Englander, J. Impact of Using Analytic Derivatives In Optimization For N-Impulse Orbit Transfer Problems. J Astronaut Sci 69, 218–250 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Two-body problem
  • Analytic derivatives
  • Orbit transfer optimization
  • Trajectory optimization
  • Impulsive maneuvers