Geometric Approach to the Perpendicular Thrust Case for Trajectory Optimization

For the case of impulsive thrust trajectories, Lawden’s primer vector theory gives a set of necessary conditions that determines if intermediate impulses have to be applied in order to obtain a fuel optimal trajectory. In this paper, a novel approach is presented in which, through the representation of the primer vector in polar coordinates, a separation of the in-plane and out-of-plane components occurs. This procedure gives a complete analytic solution for the out-of-plane component of the primer vector, which is shown to be independent of the semimajor axis of the transfer orbit. In the case where the initial and final thrusts are both perpendicular to the orbital plane, the optimality of the transfer arc is fully analyzed. The analytic correlations between the boundary conditions on the transfer orbit and the profile of the primer vector are derived. In particular, the novel approach allows the development of a simple procedure based on a graphical representation from which, given only the initial and final position vectors, the optimality of the transfer orbit can be determined