  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
multi-rendezvous 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.
