Optimal low-thrust GTO–GSO transfers using differential evolution

Abstract

Transfers from geosynchronous transfer orbits (GTO) to geosynchronous orbits (GSO) using electric propulsion have been optimized in the current study. Both time-optimal and fuel-optimal trajectories are generated. Three-dimensional equations of motion are considered for the system dynamics. The indirect approach based on optimal control theory is followed and the resulting two-point boundary value problem is solved using differential evolution, a search-based global optimization technique. Optimal trajectories for various mission scenarios are obtained using differential evolution and compared to transfers presented in literature.

List of symbols

DE:

Differential evolution

TPBVP:

Two-point boundary value problem

GTO:

Geosynchronous transfer orbit

GSO:

Geosynchronous orbit

CR:

Crossover ratio

F:

Mutation factor

NP:

Population size

$\mu _p$ :

Gravitational parameter of the] primary body (Earth)

$\mu _s$ :

Gravitational parameter of the secondary body

r :

Radial distance from the centre of the Earth

$r_s$ :

Radial distance from the centre of the secondary body

${\vec {\alpha }}$ :

Acceleration vector due to thrust

m :

Spacecraft mass

$I_{sp}$ :

Specific impulse

${\vec{\mathbf {x}}}$ :

State vector

${\vec {\lambda} }$ :

Costate vector

$\Phi _f$ :

Terminal cost function

$J_{time}$ :

Time-optimal cost function

$J_{fuel}$ :

Fuel-optimal cost function

${\hat{h}}$ :

Angular momentum unit vector

${\vec{\mathbf {e}}}$ :

Eccentricity vector

$T_{max}$ :

Maximum available thrust

$\varvec{H_{time}}$ :

Time-optimal Hamiltonian

$\varvec{H_{fuel}}$ :

Fuel-optimal Hamiltonian

${\vec {\alpha }}^*$ :

Optimal acceleration vector