The properites of the optimal and sub-optimal solutions to multiple-pass aeroassisted plane change were previously studied in terms of the trajectory variables. The solutions show the strong orbital nature. Then, it is proposed to obtain the variational equations of the orbital elements. We shall use these equations and the approximate control derived in Vinh and Ma (1990) to calculate the trajectories. In this respect, the approximate control law and the transversality condition are transformed in terms of the orbital elements. Following the above results, we can reduce the computational task by further simplification. Within omega and Omega being small and returning to the value of zero after each revolution, we neglect the equations for omega, and Omega. Also, since omega approximately equal to 0, that is alpha approximately equal to f, we can neglect the equation for the alpha and have only three state equations for the integration. Still the computation over several revolutions is long since it is performed using the eccentric anomaly along the osculating orbit as the independent variable. Here, we shall use the method of averaging as applied to the problem of orbit contraction to solve the problem of optimal plane change. This will lead to the integration of a reduced set of two nonlinear equations.
Journal of The Astronautical Sciences 45(3), pp.329-347