Optimal explicit guidance for three-dimensional launch trajectory

Abstract

The paper presents an exoatmospheric near-optimal explicit guidance algorithm for a satellite launch vehicle following a two- or three-dimensional trajectory. The unified steering law has been developed in a vector form using an equivalent gravitational field in an inertial coordinate frame. The resulting steering law follows a linear tangent law in a canted plane for a three-dimensional trajectory. The principle is based on the concepts of optimal transfer between non-coplanar orbits. The guidance parameters are determined by solving three simultaneous algebraic equations envolving thrust integrals which are computed recursively. Two constant gravity related vectors are defined which account for change in velocity and position due to the spherical Earth. The solution is obtained by a differential correction method after finding the required partial derivatives analytically. A predictor corrector approach is suggested where the trajectory is predicted using the Encke's method. This also enables consideration of other effects such as oblateness. The simulation results for typical multistage launch vehicles show that the guidance algorithm is extremely accurate, robust and yet simple enough for on-board implementation. It can be used for a variety of missions.