Numerical computation of optimal trajectories for coplanar, aeroassisted orbital transfer

This paper is concerned with the problem of the optimal coplanar aeroassist ed orbital transfer of a spacecraft from a high Earth orbit to a low Earth orbit. It is assumed that the initial and final orbits are circular and tha t the gravitational field is central and is governed by the inverse square law. The whole trajectory is assumed to consist of two impulsive velocity c hanges at the begin and end of one interior atmospheric subarc, where the v ehicle is controlled via the lift coefficient. The problem is reduced to the atmospheric part of the trajectory, thus arri ving at an optimal control problem with free final time and lift coefficien t as the only (bounded) control variable. For this problem, the necessary c onditions of optimal control theory are derived. Applying multiple shooting techniques, two trajectories with different control structures are compute d. The first trajectory is characterized by a lift coefficient at its minim um value during the whole atmospheric pass. For the second trajectory, an o ptimal control history with a boundary subarc followed by a free subarc is chosen. It turns out, that this second trajectory satisfies the minimum pri nciple, whereas the first one fails to satisfy this necessary condition; ne vertheless, the characteristic velocities of the two trajectories differ on ly in the sixth significant digit. In the second part of the paper, the assumption of impulsive velocity chang es is dropped. Instead, a more realistic modeling with two finite-thrust su barcs in the nonatmospheric part of the trajectory is considered. The resul ting optimal control problem now describes the whole maneuver including the nonatmospheric parts. It contains as control variables the thrust, thrust angle, and lift coefficient. Further, the mass of the vehicle is treated as an additional state variable. For this optimal control problem, numerical solutions are presented. They are compared with the solutions of the impuls ive model.