Solution Space of Minimum-Time Low-Thrust Rephasing in Elliptical Orbit

Exploring the solution space of indirect methods can provide initial guesses and evaluate the solutions with arbitrary design parameters. In this article, the solution space of the minimum-time low-thrust rephasing in an elliptical orbit is explored by reducing the number of design parameters to two and depicting the solution space on a set of contour maps. First, considering that the spacecraft in a general elliptical orbit rendezvous with a target at a different along-track position, the scaling and linearizing techniques are employed to obtain a set of unified linearized equations of motion. Analytical integrals of Euler–Lagrange equations are derived. Then, the boundary condition with free initial true longitude ($L_{0}$) is focused on, and some symmetry properties can be applied to simplify the high-dimensional shooting function and to obtain two locally optimal solutions. Finally, the solution space of free-$L_{0}$ problem is sufficiently explored by solving univariate monotone shooting functions with high efficiency and global optimality, and the nonlinear solutions are numerically obtained and compared with the linear solutions. A cubic interpolation method is proposed to provide initial guesses for the general fixed-$L_{0}$ problem. Numerical tests show the applications of the proposed solution space for solving nonlinear problems in two-body and high-fidelity dynamic models.