Extremal Control and Guidance Solutions for Orbital Transfer with Intermediate Thrust

Mayer’s variational problem of determining optimal trajectories of motion with constant exhaust velocity and limited mass-flow rate in a Newtonian field is considered. Analytical solutions for extremal transfers between two coplanar elliptical orbits via one and two intermediate-thrust arcs are presented. It is shown that in the cases of transfers via one and two intermediate-thrust arcs, the orbital parameters have to satisfy respectively two constraints and one constraint imposed by the continuity conditions formed at the junctions. This means that respectively two and one of these orbital parameters must not be arbitrary. Furthermore, by utilizing the analytical solutions for the intermediate-thrust arcs, the corresponding extremal guidance laws for a transfer between two points with arbitrary position vectors and for a transfer between elliptical orbits are presented. The guidance commands are formulated in terms of commanded thrust angle and commanded mass-flow rate. A numerical example illustrating the proposed trajectory and guidance solutions is discussed. The results of the computations are compared to those of similar one- and two- impulsive transfers, and a low-thrust orbit raising maneuver. It is demonstrated that for some eccentricities of the terminal orbits, the fuel consumption for a transfer via an intermediate-thrust arc is comparable to that of the two-impulsive transfers. Analyses show that a maneuver with such an arc may be fuel-efficient with longer duration and higher thrust than that of the low-thrust maneuver. The trajectory computations and guidance performances are illustrated by graphical relationships.