A FUEL-OPTIMAL transfer trajectory design problem from a low-Earth orbit to a low-lunar orbit is considered. A variable specific impulse type of low thruster is assumed as the main thruster for the transfer. Among various trajectory optimization approaches, this Note focuses on an indirect method based on the variational principle, through which the optimization problems are converted into a two-point boundary value problem (TPBVP). The trajectory optimization problem can be solved by seeking unknown initial costate variables; however, the indirect method is known to suffer from some critical drawbacks such as its small radius of convergence. Therefore, it is important to provide an accurate initial guess set for the unknown initial costates. In [1–8], an initial guess set was generated by indirect optimization procedures, which comprise several subproblems for an interplanetary transfer trajectory optimization problem. An interplanetary transfer trajectory constructed with a low-thrust input can be divided into two spiral transfer phases and a phase linking two spiral transfer trajectories (e.g., an Earth-centered spiral transfer phase, a translunar phase, and amoon-centered spiral transfer phase); therefore, the subproblems can be categorized into two parts. The first part involves problems to optimize the spiral transfer trajectories with respect to the Earth or a target planet. The second part is composed of problems to connect two spiral transfer trajectories. For the problems in the second part, equations of spacecraft motion defined in rotating frames have been used, because the motions of main massive bodies (planets) and the two spiral trajectories designed in the first part could be efficiently described concurrently [2–8]. Therefore, spiral transfer trajectories under the governing equations that include the rotational effect of the frames are required. The spiral transfer trajectories were designed by solving specific energy-targeting problems using an indirect approach. In [1], specific energy-targeting problems were considered under two-body dynamics in the inertial frame. The problems were solved using initial costate estimation methods based on a combination of the adjoint control transformation technique, functional approximation, and extrapolation. In the cases of Earth-to-Mars transfers [2–5], because the third-body perturbation and the rotational effect of the frame did not significantly alter the spirals found in the inertial frame, the initial costates in the inertial frame could be directly used as initial guesses for the specific energy-targeting problem in the rotating frame. However, in the case of the Earth-to-moon transfer [8], the effect of the rotation and the third-body perturbation significantly alter the spiral trajectories found in the inertial frame. If the initial costates found in the inertial frame are used as initial guesses for the problem defined in the Earth–moon rotating frame, the initial costates do not converge. In [8], using a homotopy process by multiplying a scale factor from zero to one with the terms due to rotational effect of the frame and the third-body perturbation, the spiral trajectory was optimized in the Earth–moon rotating frame. However, if the initial radius of the circular orbit or transfer time, or if target-specific energy is changed, additional steps, such as determining a new curve fitting, extrapolation, or a homotopy process, may be necessary to design one spiral transfer trajectory. In this Note, to reduce computational burden, an initial costate estimation method is suggested to construct spiral trajectories directly in the Earth–moon rotating frame for an arbitrary initial radius and transfer time and for arbitrary target-specific energy. In the new estimation method, an initial guess structure is generated, which is derived according to the initial costate properties. Also, as an application of the structure of the initial guess, a means of determining the duration of a spiral phase and the target-specific energy at the end of the spiral phase is suggested for Earth-to-moon transfer trajectories. This has been considered in previous papers. In the following sections, a fuel-optimal Earth-to-moon trajectory design problem is addressed. In addition, the new initial costate estimation method is presented for designing spiral transfer trajectories. Finally, fuel-optimal Earth-to-moon trajectories from a 315 km Earth-parking orbit to a 100 km lunar-parking orbit are designed using the initial guess structure and the proposed design procedure.
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