A Real-Time Approximate Optimal Guidance Law for Flight in a Plane

This paper presents a real-time guidance scheme for the problem of maximizing 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. The flight is constrained to a path in the equatorial plane while reaching an orbital altitude at orbital injection speeds. It is shown that the dynamics of the problem can be separated into primary and perturbation effects by a small parameter ε, where ε is the ratio of the atmospheric scale height to the radius of the Earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (ε = 0) can be obtained in closed form. The neglected perturbation terms are induded in the higher-order terms of the expansion. These higher-order terms are determined from the solution of first-order linear partial differential equations requing only integrations which are quadratures. These quadratures can be performed rapidly with the emerging computer capabilty so that real-time approximate optimization can be used to construct the launch guidance law. The application of this technique to flight in three-dimensions is made apparent from the solution presented here.