Time-Optimal Control of Axi-Symmetric Rigid Spacecraft Using Two Controls

In this paper, we consider the minimum-time reorientation problem of an axi-symmetric rigid spacecraft with two independent control torques mounted perpendicular to the spacecraft symmetry axis. The objective is to reorient the spacecraft from an initial attitude, with some angular velocity, to a nal attitude with a certain angular velocity in minimum time. All possible control structures, including both singular and nonsingular arcs, are studied completely by deriving the corresponding formulae and the necessary optimality conditions. It is shown that a second order singular control can be part of the optimal trajectory. It is also shown that for an inertially symmetric and a nonspinning axi-symmetric rigid body, it is possible for in nite-order singular controls to be part of or the whole optimal trajectory. In particular, for a non-spinning axi-symmetric rigid body, the secondorder singular trajectory is shown to be an eigenaxis rotation. An e cient method for numerically solving the optimal control problem, based on a cascaded computational scheme which uses both a direct method and an indirect method, is also presented. Numerical examples demonstrate optimal reorientation maneuvers with both nonsingular and singular subarcs, and comparisons are made between eigenaxis rotations and the true time-optimal rotations.