Analytical shaping method for low-thrust rendezvous trajectory using cubic spline functions

Preliminary mission design requires an efficient and accurate approximation to the low-thrust rendezvous trajectories, which might be generally three-dimensional and involve multiple revolutions. In this paper, a new shaping method using cubic spline functions is developed for the analytical approximation, which shows advantages in the optimality and computational efficiency. The rendezvous constraints on the boundary states and transfer time are all satisfied analytically, under the assumption that the boundary conditions and segment numbers of cubic spline functions are designated in advance. Two specific shapes are then formulated according to whether they have free optimization parameters. The shape without free parameters provides an efficient and robust estimation, while the other one allows a subsequent optimization for the satisfaction of additional constraints such as the constraint on the thrust magnitude. Applications of the proposed method in combination with the particle swarm optimization algorithm are discussed through two typical interplanetary rendezvous missions, that is, an inclined multi-revolution trajectory from the Earth to asteroid Dionysus and a multirendezvous trajectory of sample return. Simulation examples show that the proposed method is superior to existing methods in terms of providing good estimation for the global search and generating suitable initial guess for the subsequent trajectory optimization.