Optimal Trajectories

Optimal trajectories are analysed, covering both constant- and variable-specific-impulse cases. Primer vector is defined and illustrated. The first-order necessary conditions for an optimal constant-specific-impulse (CSI) trajectory were first derived by Lawden using classical Calculus of Variations. Variable-specific-impulse rocket engines are discussed with the cost functional for a VSI engine. In the derivation that follows, an Optimal Control Theory formulation is used, but the derivation is similar to that of Lawden. One difference is that the mass is not defined as a state variable, but is kept track of indirectly.

Improving a Nonoptimal Impulsive Trajectory

Improving a nonoptimal trajectory is analysed, including adding terminal coasts and midcourse impulses in fixed-time trajectories. Orbit transfer is also analysed. If the primer vector evaluated along an impulsive trajectory fails to satisfy the necessary conditions (NC) for an optimal solution, the way in which the NC are violated provides information that can lead to a solution that does satisfy the NC. The necessary gradients were first derived by Lion and Handelsman.

Symmetry Properties of Optimal Relative Orbit Trajectories

The determination of minimum-fuel or minimum-time relative orbit trajectories represents a classical topic in astrodynamics. This work illustrates some symmetry properties that hold for optimal relative paths and can considerably simplify their determination. The existence of symmetry properties is demonstrated in the presence of certain boundary conditions for the problems of interest, described by the linear Euler-Hill-Clohessy-Wiltshire equations of relative motion. With regard to minimum-fuel paths, the primer vector theory predicts the existence of several powered phases, divided by coast arcs. In general, the optimal thrust sequence and duration depend on the time evolution of the switching function. In contrast, a minimum-time trajectory is composed of a single continuous-thrust phase. The first symmetry property concerns minimum-fuel and minimum-time orbit paths, both in two and in three dimensions. The second symmetry property regards minimum-fuel relative trajectories. Several examples illustrate the usefulness of these properties in determining minimum-time and minimum-fuel relative paths.