A technique used to develop a real-time guidance scheme for the Advanced Launch System is presented. The present approach is to construct an optimal guidance law based upon an asymptotic expansion associated with small physical parameters, ∈. The problem is to maximize the payload into orbit subject to the equations of motion of a rocket over a non-rotating spherical Earth. The dynamics of this problem can be separated into primary effects due to thrust and gravitational forces and perturbation effects that 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 zero-order term (∈=0) can be obtained in closed form. The neglected perturbation terms are included in the higher order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only integrations that are quadratures. Improvement to the vacuum zero-order trajectory is achieved by approximating the aerodynamic perturbation effect as functions of the independent variable over subarc intervals. The results of the expansion method are presented and compared to a numerical optimization scheme and to an expansion of the Euler-Lagrange first-order optimality conditions.
