Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem

Abstract

Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the doublehomotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to an Earth-Moon L 1 halo orbit is presented to substantiate the effectiveness of the technique. INDEX TERMS Circular restricted three-body problem, Earth-Moon transfers, homotopy method, trajectory optimization.

Advances in Trajectory Optimization for Space Vehicle Control

“…A promising modern indirect method family relies on homotopy in order to solve the TPBVP (Pan et al, 2016(Pan et al, , 2019Pan and Pan, 2020;Taheri and Junkins, 2019;Trélat, 2012;Rasotto et al, 2015;Lizia et al, 2014). Homotopy aims to address the aforementioned challenges of slow convergence, initial guess quality, and active constraint specification.…”

“…Numerous solution methods have been studied, including methods based on primer vector theory (Russell, 2007;Petropoulos and Russell, 2008;Restrepo and Russell, 2017), direct methods based on solving an NLP (Betts, 2000;Arrieta-Camacho and Biegler, 2005;Ross et al, 2007;Starek and Kolmanovsky, 2012;Rao, 2015, 2016), and indirect methods (Alfano and Thorne, 1994;Fernandes, 1995;Kechichian, 1995;Haberkorn et al, 2004;Gong et al, 2008;Gil-Fernandez and Gomez-Tierno, 2010;Zimmer et al, 2010;Pan et al, 2012;Pontani and Conway, 2013;Cerf, 2016;Taheri et al, 2016Taheri et al, , 2017Rasotto et al, 2015;Lizia et al, 2014). Some recent advances for indirect methods include homotopy methods (Pan et al, 2019;Pan and Pan, 2020;Cerf et al, 2011), optimal switching surfaces (Taheri and Junkins, 2019), the RASHS and CSC approaches from Section 2.5.2 (Saranathan and Grant, 2018;Taheri et al, 2020a,b), and simultaneous optimization (also known as co-optimization) of the trajectory and the spacecraft design parameters (Arya et al, 2020).…”

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

“…Among the OCPs employed for space exploration missions, minimum time trajectories and fuel optimal trajectories, which are also considered in this paper, play a key role especially during the orbit transfer phases and the proximity maneuvers around a target body. Applications of such optimal trajectories already studied in the literature include cislunar trajectories [7][8][9][10][11], low−thrust orbit transfer around the Earth [12], interplanetary trajectories [13], and solar sail optimal trajectories [14]. Specifically, the primary problem centers on reducing propellant consumption (or maximizing final mass), whereas the secondary objective is to minimize the time of flight.…”

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

“…In [21,31,32], the fuel-optimal bang-bang control was approached by a sequence of continuous controls. Moreover, pseudo-arclength method [30], double-homotopy method [33], bounding homotopy method [34] and TFCbased homotopy method [35] have been explored to tackle failures of the continuation process. The solution quality is also closely linked to the homotopy method.…”