A Smoothed Eclipse Model for Solar Electric Propulsion Trajectory Optimization

Abstract

Solar electric propulsion (SEP) is the dominant design option for employing low-thrust propulsion on a space mission. Spacecraft solar arrays power the SEP system but are subject to blackout periods during solar eclipse conditions. Discontinuity in power available to the spacecraft must be accounted for in trajectory optimization, but gradient-based methods require a differentiable power model. This work presents a power model that smooths the eclipse transition from total eclipse to total sunlight with a logistic function. Example trajectories are computed with differential dynamic programming, a second-order gradient-based method.

Optimal Space Trajectories with Multiple Coast Arcs Using Modified Equinoctial Elements

“…Most recently, Mazzini [33] utilized a smooth eclipse function, to restore the regularity conditions, for the purpose of applying the Pontryagin minimum principle, whereas Taheri and Junkins [50] proposed the hyperbolic tangent as a smoothing function. Similarly, Aziz et al [1] smoothed the eclipse transition with a logistic function and computed the optimal paths through differential dynamic programming, a second-order gradient-based method.…”

Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach

“…This is particularly relevant since most of the existing EP system are solar powered and therefore they are unable to operate during solar eclipse periods. Indeed, as pointed out in [21,22], low-thrust trajectory optimization without concerns for eclipsing is futile when subsequent analysis reveal that a thrusting maneuver has been scheduled in the shadow region. The net effect of solar eclipses is to introduce a state-dependent discontinuity in the control input structure, which constitutes a serious challenge for gradient-based optimization techniques.…”

“…An advantage of this method over multi-phase formulations is that it requires no prior information about the structure of the optimal solution, although some kind of smoothing is needed for the system dynamics. In [22], a single-phase approach is employed in combination with a smoothed eclipse model in order to solve a minimumfuel problem involving a large number of orbital revolutions. This is done by using hybrid differential dynamic programming [25].…”

“…The aim of this paper is to present a new approach for the optimization of low-thrust orbit transfers, employing a direct collocation method. The work is in the same spirit of [22,26], but some key features are introduced in the problem definition. First, equinoctial elements are replaced by a novel nonsingular set of ideal elements inspired by Hansen's theory (see [27][28][29]) for the description of the orbital motion.…”

“…The adopted cost function involves a convex combination of TOF and fuel consumption, providing a suitable way to trade off these objectives by means of a scalar parameter. Eclipse effects are taken into account by adapting the smoothing technique in [22] to the the proposed state parametrization. The OCP is transcribed into a sparse NLP by using the commercial package GPOPS-II [32], which implements an hp-adaptive Legendre-Gauss-Radau pseudospectral collocation strategy.…”