An analysis of first-order singular thrust-arcs in rocket trajectory optimization

Abstract

The time-free, Mayer-optimal control problem for coplanar rocket flight in an atmosphere is formulated. Lie brackets are used to analyse the resulting singular thrust magnitude and steering along the singular arc. Previously, an analysis of the endoatmospheric singular arc had been a source of frustration due to the laborious mathematical manipulations involved and the lack of necessary integrals to eliminate the annoying costate variables. While the Lie-bracket formalism does not appreciably reduce the mathematical labor, it is readily apparent by the choice of polar coordinates that sufficient integrals of motion exist to cast the singular control in a state feedback form for both optimal and prescribed, fixed-angle steering. From this result, it is shown that non-optimal steering alters the structure of the singular arc. Consequently, a steering-induced, first-order, exoatmospheric singular arc is derived by considering the limiting case of a vanishing atmosphere. For the special case of steering along the velocity vector, it is shown that the exoatmospheric singular thrust magnitude is simply equal to a component of the gravitational force.