Real-Time Approximate Optimal Guidance Laws for the Advanced Launch System

An approach to optimal ascent guidance for a launch vehicle is developed based upon an expansion technique. The problem is to maximize payload into orbit subject to the equations of motion of a rocket over a rotating spherical earth. It is assumed that the thrust and gravitational forces dominate over the aerodynamic forces. It is shown that these forces can be separated by a small parameter ¿, where ¿ is the ratio of the atmospheric scale height to radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in a series where the zeroth-order term (¿=0) can be obtained in closed form. The zeroth-order problem is that of maximum payload into orbit subject to the equations of motion of a rocket in a vacuum over a flat earth. The neglected inertial and aerodynamic terms are included in higher-order terms of the expansion. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only quadrature integrations. These quadrature integrations can be performed rapidly with the emerging computer capability so that real-time approximate optimization can be used to construct the launch guidance law.