Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient. The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit. In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.