WITHDRAWN: On the Pointing-Constrained Fuel-Optimal Powered Descent Problem

This paper presents theoretical advancements to the three degree-of-freedom (3DOF) powered descent guidance (PDG) problem. A common constraint in the 3DOF PDG problem is the pointing constraint, i.e., an angular limit on the direction of the thrust steering vector constraining it to lie within a cone. New insights are gained into this control-only inequality constraint by analyzing the pointing constraint switching function, whose sign governs its activation. It is shown this constraint will become active at most once on extremals. Moreover, the switch time, should it occur, from active to inactive (or inactive to active) arcs on admissible solutions can be calculated in closed-form. Closed-form solutions to the state and costate ordinary differential equations (including the mass costate) are also derived when this constraint is active. These results can be used to extend fast guidance algorithms based on solving the piecewise algebraic root finding problem, resulting from the necessary conditions of optimality.