Optimal Control of Transfer to Vertical Orbits from Lyapunov Orbits Using Low-Thrust Engine

There are many different families of periodic orbits in the Earth-Moon system, such as Lyapunov orbits, halo orbits, vertical orbits, etc. The establishment of a lunar space station requires a spacecraft to be able to transfer among these orbits. Lyapunov orbits have been used by some missions and are well-studied orbits, while periodic vertical orbits can provide large amplitudes of spacecraft motion outside the plane of the Moon’s motion, which makes it possible to avoid shadowing of the orbits and use them as relay satellite in cislunar space. Modern researchers mainly consider the use of high-thrust engines for transfer. With the development of electric propulsion technology, the use of low, but long-acting thrust for deep space exploration has become especially relevant. This is due to the high specific characteristics of the propulsion systems of this type. In this article, an algorithm has been developed for determining the optimal control with a low-thrust engine for a transfer from Lyapunov orbit to a vertical orbit. The minimum time of flight or the minimum costs of the working body are used as criteria for optimality. In the calculation for solving the two-point boundary value problem of the optimal control theory, the parameter continuation algorithm is used, which allows to gradually get the transfer from some simple results to the final transfer trajectory. The results obtained make it possible to assert that the use of intermediate axial orbits allows the use of propulsion systems with lower thrust levels. In this case, the duration of the flight increases slightly with an almost unchanged consumption of the working body. Moreover, the homotopy method makes it possible to reduce the consumption of working body, while the control of the engine throttling becomes discrete. The results of this study and the algorithms proposed in this article can be used to determine the optimal program control and ballistic design of lunar missions.

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