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 p→i :
-
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
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This paper is based upon work supported by NASA, Award Number 80NSSC19K1642
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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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DOI: https://doi.org/10.1007/s40295-022-00318-y