Primer vector theory based optimal guidance for orbital transfers in cis-lunar environment
Authors:   University of Pisa, Pisa, Italia, Gemignani Gabriele , Innocenti Mario , Bucchioni Giordana 
Published: January 2024

Over the past years, the commercial interest in returning to the Moon and in exploring the deep space is increasing, this making the research in the field prosperous. In this context, Spacecraft Trajectory Design covers a fundamental role in the success of the overall mission. Fuel minimization is a key factor in the design of space manoeuvres, due to the desire of improving the efficiency of future missions to the Moon and to the outer space.
The proposed work is inserted in this context, since it aims to provide a full optimization algorithm for fixed-time transfer trajectories between non-Keplerian orbits in the Earth-Moon framework, with the objective of minimizing the total propellent usage in terms of impulsive manoeuvres. The dynamics of Circular Restricted Three Body Problem (CR3BP) is used as mathematical model, as the majority of the state-of-the-art works present in literature considers it as a valuable approximation for preliminary studies in mission design analysis.
The method is based on Primer Vector Theory, which gives the necessary conditions of local optimality and provides a clear indication of where to add intermediate impulses in order to decrease the total cost. The algorithm automatically generates the optimal sequence of impulses for a given transfer to occur at the lower cost possible. The procedure is mainly composed by three steps: first, a 2-impulses initial trajectory is guessed, then a midcourse impulse is initialized with Primer Vector theory, thanks to the investigation of the Primer Vector along the trajectory, and finally its position and time are optimized via a minimization procedure. The whole sequence is repeated until the transfer path satisfies the Primer Vector optimality conditions.
In this work, the initial guess is computed via “orbit chaining”, a technique that selects a fixed number of intermediate points located on orbits belonging to the same family and then interpolates them. The Time of Flight is computed in this step and kept fixed for the whole optimization process. The position and time of all the intermediate impulses are optimized with a Broyden–Fletcher–Goldfarb–Shanno algorithm, which exploits the knowledge of the gradient of the cost function, namely the sum of the modules of Δv. The non-linear nature of the problem implies the existence of some numerical issues, which are also discussed in this work. The most significant ones are the singularities of the model in the proximity of the attractive bodies and the convergence problem during the computation of arcs, which is related to the Multiple shooting procedure.
The algorithm is applied for two specific orbits around Lagrange point L2, which is one of the five equilibrium points of CR3BP. More specifically, Near Rectilinear Halo Orbits are considered, a class of non-Keplerian orbits which are periodic and weakly unstable. Results are shown, in terms of total Δv required and optimality achievness, for all the combinations of transfer from 36 departure states to 36 arrival states, each one identified by the non-Keplerian equivalent of the mean anomaly on the corresponding orbit. From the results shown, the proposed method highlights good performance in terms of Δv saving and computational complexity.