Faster Low-Thrust Trajectory Design Through Finite Fourier Series

Abstract

Low-thrust propulsion has gained more popularity over the past few decades because of its high efficiency. Interplanetary transfer trajectories in particular benefit from low-thrust propulsion, considering the typically high Delta V to be achieved. In order to allow a fast design of such missions, first-order, efficient representations of transfer orbits are usually used before a more detailed and exact numerical model is applied. The finite Fourier series method is one of these so-called shape-based methods that are suited for this. In this thesis, the focus is twofold: it lies on the implementation and validation of the method in TUDAT, and on improving this first-order method through a different initialisation strategy with the goal of improving its convergence speed and its three-dimensional stability.

During implementation, some inconsistencies have been encountered that were not clearly, or even incorrectly addressed or documented by the inventors of this method. These include the calculation of the initial guess, the definition of the decision vector, the performance of the two-dimensional unconstrained finite Fourier series, the two-dimensional reference results and the interpretation of the reference frame. After these problems have been overcome, the method has been validated successfully against three case studies.

The original strategy is based on the approximation of the trajectory by means of a third-order power function. In the search of a better strategy, four different function types have been analysed: (the original) power function, an exponential function, a trigonometric function and a logarithmic function. The functions were tested on two transfer trajectories, both in two and three dimensions: from the Earth to Jupiter and from the Earth to Dionysus.

It has been concluded that the use of the proper initialisation strategy can considerably boost the effectiveness of the finite Fourier series method. However, a clear contrast between the two-dimensional and three-dimensional version was observed: the two-dimensional version of the algorithm greatly benefits from an exponential function to generate a priori values and shows an increase in convergence speed of up to 42.6%. On the other hand, there has not been a single approach that decreases the convergence time of the solver nor improves the stability for the three-dimensional version. This also led to the conclusion that the finite Fourier series method is rather sensitive regarding a priori values for the axial candidate.