Minimum fuel spacecraft reorientation

Abstract

Fuel optimal solutions for the reorientation of an inertially symmetric rigid spacecraft with independent three-axes controls are investigated. All possible optimal control strategies are identified. These include bangbang solutions, finite-order singular arcs, and infinite-order singluar arcs. Higher order necessary conditions for optimality of finite-order singular arcs are presented. Numerical examples of fuel optimal solutions with fixed maneuver time are presented involving all of the theoretically possible control logics. I. Introduction O PTIMAL rigid body rotational maneuvers have been studied by many researchers in the context of spacecraft attitude control systems.15 Minimum time problems with bounded control energy and/or minimum control effort problems subject to fixed maneuver time have been the main objectives in these studies. In a recent study Bilimoria and Wie1 were the first to successfully solve minimum time 180-deg rest-to-rest reorientation problems subject to bounded control torques through a rigorous optimal control approach. Later, Seywald and Kumar6 gave a complete analysis of all possible control logics associated with the same dynamical system and performance index. Besides the nonsingular control solutions found in Ref. 1, finite- and infinite-order singular control strategies were derived and numerical examples involving these control logics were presented. In the present paper, the authors investigate the problem of reorienting an inertially symmetric spacecraft with minimum fuel expenditure from given initial angular position and velocity to fully or partly prescribed final positions. Three bounded independent control torques are assumed with the control axes aligned along prescribed principal axes. The mass of fuel burned is ignored in that the moments of inertia are unchanged during the maneuver. Except for an additional mass differential equation, the dynamical system used here is the same as in Refs. 1 and 6. The identification of all possible optimal control logics for the problem stated is emphasized. To avoid absolute values in the right-hand side of the fuel mass differential equation, each control torque is divided into its positive and its negative component, thus giving rise to six mathematical controls instead of the three physical controls. For each of the six mathematical controls, the possible optimal control logics are either bangbang type, or of finite/infinite-order singular type. Higher order necessary conditions for optimality of finiteorder singular arcs are examined using Goh transformations of the associated accessory minimum problem. Numerical examples of optimal solutions with bang-bang structure as well as finite- and infinite-order singular arcs are presented.