  This study proposes an analytical Delta-V approximation of short-time
transfers based on the linear relative motion and a gradient-based nonlinear
programming model of multi-target rendezvous and flyby trajectories. In
previous studies, the Lambert's solution is commonly used to evaluate Delta-V
of short-duration transfers. In this study, to avoid the iteration process for
obtaining the Lambert's solution and its gradient, the linear relative motion
equations are applied to form an analytical two-point boundary value model for
the near-circular orbit rendezvous problems. Although the relative motion
equations are usually applicable when the two orbits are close enough, and the
position and velocity errors would become more significant as the orbital
differences increase, the errors of the velocity increments were proved
acceptable in our simulations. Moreover, the analytical formula facilitates the
calculation of the gradients to the start epoch and flight time, which are used
to establish a nonlinear programming model for sequence optimization that
gradient-based algorithms can easily solve. Simulation results demonstrated
that the analytical Delta-V approximation requires much less calculation than
the Lambert's solution, and the proposed gradient-based nonlinear programming
algorithms can obtain similar results in less time than previous methods.
