Techniques for Developing Approximate Optimal Advanced Launch System Guidance

In this paper an extension to our previous technique used to develop a real-time guidance scheme for the Advanced Launch System is presented. Our approach is to construct an optimal guidance law based upon an asymptotic expansion associated with small physical parameters, ¿. The problem is still to maximize the payload into orbit subject to the equations of motion of a rocket over a nonrotating spherical Earth. The trajectory of a rocket modeled as a point mass is considered with the flight restricted to an equatorial plane while reaching an orbital altitude at orbital injection speeds. The dynamics of this problem can be separated into primary effects due to thrust and gravitational forces, and perturbation effects which include the aerodynamic forces and the remaining inertial forces. An analytic solution to the reduced-order problem represented by the primary dynamics is possible. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (¿ = 0) can be obtained in closed form. The neglected perturbation terms are included in the higher-order terms of the expansion. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only integrations which are quadratures. These quadratures can be performed rapidly with the emerging computer capability so that real-time approximate optimization can be used to construct the launch guidance law. Here ¿ is chosen as the ratio of the atmospheric scale height to the radius of the Earth. It is important that the perturbation effects remain small.