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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.

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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 :

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 :

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

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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

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Correspondence to Ahmed Ellithy.

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

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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

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Keywords

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