An investigation on Shape-based methods with different polynomial bases for Low-Thrust mission design
Creators: name:  P. SOLANO LOPEZ 
name:   S. CUEVAS DEL VALLE 
name:   A. SOLA MOLINA 
name:   A. BAREA VILLAR 
name:   H. URRUTXUA CEREIJO 
Publisher: name: Proceedings of  the 9th European Conference for Aerospace Sciences. Lille, France, 27 June - 1 July, 2022
Year: 2022
Subjects: FLIGHT DYNAMICS
GNC
AVIONICS
OPTIMIZATION

description: The pre-design of low-thrust complex missions with different orbital phases and their transfers is still a non-straightforward problematic that calls for the application of non-straightforward, computationally heavy set of mathematical techniques. Moreover, different eventualities might alter the initial planning of such missions, resulting in a reiteration of these heavy calculations. Consequently, alternative tools are being more and more studied to facilitate these kinds of analysis and calculations. Within the wide range of mathematical approaches in the state-of-the-art, shape-based methods have gained increasing attention within the astronautical community in recent times, with extensive applications within optimization problems. The most common technique has been that of the spectral polynomial bases in direct collocation as they are a well-known tool in the numerical community. However, their limitations have been reached too, calling for different perspectives. Among them, the exploitation of metaheuristic algorithms for trajectory design has introduced other polynomial families into the topic, such as Bezier functions. Despite their successful application to low-thrust trajectory design optimization in previous works, their mathematical formulation has not been completely explored, nor their possible numerical and computational advantages. This work presents a novel approach to the application of Bezier functions to these optimal problems using an alternative basis. In particular, we propose an orthonormal version of their generating Bernstein polynomials, so as to enhance their numerical behaviour and ease and augment convergence in their application. Additionally, the numerical performance of these curves within collocation methods is increased by exploiting the natural roots or nodes of Bezier curves. Up to now, previous work has focused on Bezier curves discretization and interpolation by means of non-optimal node selection techniques. Nevertheless, we show that by means of exploiting the natural formulation of these polynomials, their interpolation accuracy is not only ensured, but interestingly, the dimensionality of the problem at hand is also reduced. This will consequently provide clear advantages within standard optimization solving techniques, improving its performance. Thirdly, the use of different bases, such as that of Chebyshev, and the benefits they may bring in combination to this new methodology are investigated, generating a consistent mathematical formulation of the problem that accepts any kind of polynomial basis. The proposed algorithms and techniques are demonstrated and validated within several end-to-end low-thrust mission design cases. Specifically, low-thrust transfers to several binary asteroids and planetary systems are solved using low-order, orthonormal Bezier curves together with metaheuristic algorithms to illustrate the performance of the proposed enhancements. We then show the computational benefits of this novel formulation by confronting in the aforementioned missions against both the traditional approaches, the use of spectral polynomial bases with collocation techniques and the previously studied Bezier functions. To sum up, this work presents a consistent mathematical formulation for shape-based initial mission design with special application to low-thrust missions, being also capable to be extended for other problems such as solar-sail trajectories. Additionally, the consequent set of numerical tools that can be derived from this formulation is presented, tested, and validated to facilitate the design process and calculations in real mission environments.