ANALYTIC THEORY OF OPTIMAL PLANE CHANGE BY LOW AERODYNAMIC FORCES

The properties of the optimal and sub-optimal solutions to multiple-pa ss aeroassisted plane change have been studied in terms of the traject ory variables. The solutions show the strong orbital nature. It is pro posed to obtain the variational equations of the orbital elements. We shall use these equations and the approximate control derived from the suboptimal solution to calculate the trajectories. In this respect, t he approximate control law and the transversality condition are transf ormed in terms of the orbital elements. Following the above results, w e can reduce the computational task by further simplification. Since t he argument of the perigee and longitude of the ascending node are sma ll and we set their respective values to zero after each revolution, w e can neglect their equations. Also, since argument of the perigee app roximately equals zero, we can neglect the equation for the angle meas ured from line of ascending node and have only three state equations f or the integration. The computation over several revolutions is long s ince it is performed using the eccentric anomaly along the osculating orbit as the independent variable. Here, we shall use the method of av eraging as applied to the problem of orbit contraction to solve the pr oblem of optimal plane change. This will lead to the integration of a reduced set of only two nonlinear equations. The result is a mathemati cal tool for fast and accurate evaluation of the optimal plane change for a multiple-pass maneuver for a lifting re-entry vehicle.