Solving Complex Low Earth Orbit-to-Geostationary Earth Orbit Transfer Problems Using Uniform Trigonometrization Method

Low-thrust orbit transfer problems are central to reducing mission costs and enabling cleaner, more efficient space travel. However, they remain difficult to solve using mathematically superior indirect methods of optimization. This is mainly due to the sensitivity to initial guesses and ill-conditioned matrices generated using traditional indirect methods. This paper applies the Uniform Trigonometrization Method (UTM), a cutting-edge indirect optimization technique, to four cases of low-thrust low Earth orbit (LEO)-to-geostationary Earth orbit (GEO) transfer problems. Using the UTM framework, including efficient numerical continuation and problem scaling strategies, smoother optimal control solutions were obtained. The convergence of standard boundary value problem solvers, like MATLAB’s bvp4c, significantly increases while using the simplicity and efficiency of the UTM. The UTM was able to solve Case 1 in a simpler manner compared to the traditional indirect method presented in the literature. In Case 2, the UTM found results for a constant thrust value of 1 N, while a direct pseudospectral method failed to converge. The results obtained using the UTM for Case 2 have 20 times longer flight duration and revolutions of spacecraft around the Earth. The UTM efficiently performs trade studies using a continuation approach that generates additional insights into all cases of this problem. In Case 4, the UTM was able to easily generate a bang–bang control structure, which traditionally requires solving a complex multi-point boundary value problem. The results generated using the UTM are very high-resolution, as it relies on the necessary conditions of optimality and guarantees locally optimal solutions. These findings position the UTM as a promising indirect approach for solving real-world long-duration orbit transfers.