Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution

This paper explores the use of the averaging method in the optimal control problem related to the multirevolution orbital transfer of a spacecraft with low-thrust capabilities. The regularized equations of motion are expressed using modified equinoctial elements with the eccentric longitude as a fast variable. The control function is represented as a Fourier series relative to the eccentric longitude. The classical averaging technique’s usage results in the averaged trajectory depending only on a limited number of optimization parameters. Moreover, when transferring between near-circular orbits, the averaged motion can be estimated using analytical formulas. As such, the optimal multiorbit flight problem is simplified to nonlinear programming with fewer parameters, thereby accelerating the optimal solution’s derivation. Two practical examples illustrate the technique’s application: orbital transfer near the geostationary orbit and circular orbit raising maneuver. The solutions derived are compared with Pontryagin extremals.