Minimum Time-Energy Pull-up Maneuvers for Airborne Launch Vehicles

In this paper, the minimum time-energy pull-up maneuver problem for airborne launch vehicles (ALV) is studied with a focus on developing a numerical approach for solving the problem. Firstly, the six-degree-of-freedom (6DOF) dynamics for the motion of the ALV subject to the aerodynamic forces, the gravitational force, the propulsive force and the path constraints are established. Then, first-order necessary conditions are derived by applying the Pontryagin Maximum Principle, and the optimal control problem is transformed into a two-boundary value problem, which is generally solved numerically thanks to a shooting method. However, the convergence domain of the shooting method is very small due to high dimension and to nonlinear coupling of attitude and trajectory motions.To overcome this difficulty, we design an algorithm combining the multiple shooting method and the Predictor-Corrector continuation (PC continuation) method, where the choice of homotopy parameters relies on a careful analysis of the nature of the dynamics.Numerical results presented for pull-up maneuvers of an ALV show that the algorithm is efficient and robust with respect to terminal conditions. Our method is also applied to the problem of rapid maneuver of the upper stage of a launch vehicle (LV)