Analytic Theory of Optimal Plane Change by Low Aerodynamic Forces

Abstract

The properties of the optimal and sub-optimal solutions to multiple-pass aeroassisted plane change have been studied in terms of the trajectory variables. The solutions show the strong orbital nature. It is proposed to obtain the variational equations of the orbital elements. We shall use these equations and the approximate control derived from the sub-optimal solution 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. Since the argument of the perigee and longitude of the ascending node are small and we set their respective values to zero after each revolution, we can neglect their equations. Also, since argument of the perigee approximately equals zero, we can neglect the equation for the angle measured from line of ascending node and have only three state equations for the integration. 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 only two nonlinear equations. The result is a mathematical tool for fast and accurate evaluation of the optimal plane change for a multiple-pass maneuver for a lifting re-entry vehicle.