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

## Abstract

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.

## Abbreviations

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 :

Δ 𝜃 :

difference between two true anomalies, rad

E :

H :

n :

e :

eccentricity

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

1,2,3:

point on orbit

−:

before impulse

+:

after impulse

## References

1. Abdelkhalik, O., Mortari, D.: N-impulse orbit transfer using genetic algorithms. J. Spacecr. Rocket. 44(2), 456–460 (2007). https://doi.org/10.2514/1.24701

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)

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). https://doi.org/10.2514/1.G001030

4. Battin, R.H.: An introduction to the mathematics and methods of astrodynamics, rev. edn. American Institute of Aeronautics and Astronautics (1999). https://doi.org/10.2514/4.861543

5. Colasurdo, G., Pastrone, D.: Indirect optimization method for impulsive transfers. In: Astrodynamics Conference, AIAA 1994-3762 (1994). https://doi.org/10.2514/6.1994-3762

6. Conway, B.: Spacecraft trajectory optimization. Cambridge University Press (2009). https://doi.org/10.1017/cbo9780511778025

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). https://doi.org/10.2514/1.g003077

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). https://doi.org/10.2514/1.a34946

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). https://doi.org/10.2514/3.21662

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). https://doi.org/10.1145/2168773.2168774

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). https://doi.org/10.1016/j.actaastro.2010.02.014

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). https://doi.org/10.2514/6.2000-689

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

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). https://doi.org/10.2514/1.g001920

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). https://doi.org/10.2514/1.55592

18. Prussing, J.: A class of optimal two-impulse rendezvous using multiple-revolution Lambert solutions. J. Astronaut. Sci. 48(2-3), 131–148 (2000). https://doi.org/10.1007/bf03546273

19. Prussing, J., Chiu, J.H.: Optimal multiple-impulse time-fixed rendezvous between circular orbits. In: Astrodynamics Conference, AIAA 1984-2036 (1984). https://doi.org/10.2514/6.1984-2036

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). https://doi.org/10.1007/s40295-015-0073-x

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). https://doi.org/10.1016/j.paerosci.2018.07.007

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)

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). https://doi.org/10.1155/2015/949480

## Acknowledgements

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

## Funding

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

## Author information

Authors

### Corresponding author

Correspondence to Ahmed Ellithy.

## Ethics declarations

### Conflict of Interests

The authors declare that they have no conflict of interest.

### 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

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). https://doi.org/10.1007/s40295-022-00318-y

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s40295-022-00318-y

### Keywords

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