Indirect Optimization of Fuel-Optimal Many-Revolution Low-Thrust Transfers With Eclipses

Abstract

An efficient indirect method is presented to determine fuel-optimal many-revolution low-thrust transfers in presence of Earth-shadow eclipses. Specifically, the events of shadow entrance and exit are modelled as interior-point constraints. Following the observation that an ill-conditioned state transition matrix may occur when the spacecraft flies over the edge of the shadow, a two-level continuation scheme is introduced to generate many-revolution trajectories. The established computational framework integrates analytic derivatives, switching detection and continuation with an augmented flowchart, which yields discontinuous bang-bang solutions and their gradients. Transfers from a geostationary transfer orbit to a geostationary orbit are simulated to illustrate the effectiveness and efficiency of the method developed.

An Indirect Formulation of Operational Compliant Low-thrust Trajectories

“…( 18) the detection of the switching time 𝑡 𝑠 is essential because discontinuities cause accumulation of the integration error around those instants, and because it is necessary to introduce the compensation matrices 𝚿(𝑡 𝑠 ) through Eq. (20). However, no prior information on the structure of the thrusting profile is known.…”

“…This is particularly true as the number of coasting arcs to be imposed increases and their duration decreases. For this reason, a triple continuation scheme based on the eclipses imposition in [20] is implemented here. The approach called 𝑁 𝑚𝑎𝑥 continuation, gradually turns inactive coasting arcs into active ones, i.e., the maximum number of forced coasting arcs 𝑁 𝑚𝑎𝑥 increases at each iteration of the continuation scheme.…”

“…They allow the solution of the original, difficult, and discontinuous problem, starting from an easier one [16]. It is usually employed in indirect methods to overcome discontinuity problems as bang-bang control in fuel optimal solutions [17,18], to impose varying maximum thrust models [19] and shutdown of the thruster due to eclipses [20]. Recently, the problem of forced coasting arcs imposition has become more and more relevant, and homotopic techniques have been employed to solve it [21,22] also in direct formulations [23].…”

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

“…Regarding the quadratic perturbing function, it serves as a bridge connecting an energy-optimal problem to a fuel-optimal problem and has been utilized in numerous studies. To cite some examples, it was employed in [17] to derive low−thrust fuel−optimal trajectories, in [18] to calculate fuel−optimal low−thrust Earth−orbit transfers, accounting for shadow eclipses, and in [19] to study fuel optimal soft landing trajectories on asteroids. As an example of the logarithmic perturbing function, it is exploited by Izzo and Öztürk [20] to obtain the fuel−optimal trajectories required to build a dataset and train a Deep Neural Network (DNN) in a supervised fashion.…”

A Homotopic Direct Collocation Approach for Operational-Compliant Trajectory Design

“…Homotopy, or continuation, is a suitable method to deal with discontinuous structures. The homotopic approach allows solving the original, difficult and discontinuous problem, starting from an easier and affordable one [ 19 , 34 ]. In particular, it has been applied to indirect methods to overcome discontinuity problems as bang-bang control in fuel optimal solutions [ 3 , 11 , 35 ].…”