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Preliminary mission design requires an efficient and accurate approximationto the low-thrust rendezvous trajectories, which might be generallythree-dimensional and involve multiple revolutions. In this paper, a newshaping method using cubic spline functions is developed for the analyticalapproximation, which shows advantages in the optimality and computationalefficiency. The rendezvous constraints on the boundary states and transfer timeare all satisfied analytically, under the assumption that the boundaryconditions and segment numbers of cubic spline functions are designated inadvance. Two specific shapes are then formulated according to whether they havefree optimization parameters. The shape without free parameters provides anefficient and robust estimation, while the other one allows a subsequentoptimization for the satisfaction of additional constraints such as theconstraint on the thrust magnitude. Applications of the proposed method incombination with the particle swarm optimization algorithm are discussedthrough two typical interplanetary rendezvous missions, that is, an inclinedmulti-revolution trajectory from the Earth to asteroid Dionysus and amulti-rendezvous trajectory of sample return. Simulation examples show that theproposed method is superior to existing methods in terms of providing goodestimation for the global search and generating suitable initial guess for thesubsequent trajectory optimization.

Analytical Shaping Method for Low-Thrust Rendezvous Trajectory Using Cubic Spline Functions