A minimum-fuel mode for steady-state attitude control of an earth-orbiting spacecraft

Steady-state reaction control of the attitude of a spacecraft in the presence of environmental disturbance torques
T
is usually accomplished by using the thrusters to establish limit-cycle motion of acceptable amplitudes about all three axes. For such strategies, the greatest lower bound for the propellant consumption is proportional to
\int\min{t1}\max{t2}\|T\|dt
. The results presented in this paper show that when the disturbances are not unidirectional, the environment, consisting of gravity-gradient and aerodynamic torques, can be used to reduce this lower bound significantly. The linearized equation which describes the dynamics of an inertially referenced satellite has a periodic coefficient matrix and forcing vector. The existence of forced periodic solutions of such equations is established, and an algorithm for determining an initial point on the trajectory of these solutions in state space is provided. This defines a "natural" limit cycle, i.e., both turning points of this motion are produced by the environment. In particular, the periodic rotational motion about two of the spacecraft axes is produced by the aerodynamic torque, an effect which is usually neglected.