Introduction

A spacecraft in flight is a dynamical system. As dynamical systems go, it is comparatively straightforward; the equations of motion are continuous and deterministic, for the unforced case they are essentially integrable, and perturbations, such as the attractions of bodies other than the central body, are usually small. The difficulties arise when the complete problem, corresponding to a real space mission, is considered. For example, a complete interplanetary flight, beginning in Earth orbit and ending with insertion into Mars orbit, has complicated, time-dependent boundary conditions, straightforward equations of motion but requires coordinate transformations when the spacecraft transitions from planet-centered to heliocentric flight (and vice versa), and likely discrete changes in system states as the rocket motor is fired and the spacecraft suddenly changes velocity and mass. If low-thrust electric propulsion is used, the system is further complicated as there no longer exist integrable arcs and the decision variables, which previously were discrete quantities such as the times, magnitudes and directions of rocket-provided impulses, now also include continuous time histories, that is, of the low-thrust throttling and of the thrust pointing direction. In addition, it may be optimizing to use the low-thrust motor for finite spans of time and “coast” otherwise, with the optimal number of these thrust arcs and coast arcs a priori unknown.

Since the cost of placing a spacecraft in orbit, which is usually the first step in any trajectory, is so enormous, it is particularly important to optimize space trajectories so that a given mission can be accomplished with the lightest possible spacecraft and within the capabilities of existing (or affordable) launch vehicles.