the flexing. A rigid body model can therefore be used with the assumption that energy is dissipated without bothering to model the dissipation mechanism. This technique is called the energy sink approach. For example, it is well-known that energy dissipation in those widely oscillating appendages produced an instability in Explorer I that could hardly be called a second order effect. It was the energy dissipation, and not the vibrations of the antenna that fundamentally altered the attitude motion of Explorer I. Gravity-gradient effects on flexible satellites generate energy dissipation just as the tides on the Moon cause the Moon to move toward Earth. Therefore, in this paper, only the influence of the gravity-gradient torque on simple rigid body models with damping representing the SPS attitude will be addressed and discussed. Because the gravitational torque exists throughout the spacecraft environment, some procedure is needed to maintain the SPS nominal solar orientation. Active control using hardware such as gas jets, reaction wheels, and control moment gyros may be used to meet the needs of the mission. Nevertheless, active control for such a large system requires a great deal of power and complex hardware. Therefore, for a consumable SPS system, a major criterion on active control strategies is to use the control means as efficiently as possible. A method is introduced in the first part of this paper for achieving quasi-inertial orientation of an orbiting spacecraft with minimal control effort under the condition that no energy dissipation for the system is present. The quasi-inertial attitude mode (pendulum mode) is a motion wherein the oribiting satellites oscillate at the period of the gravity-gradient torque about the orbit normal with two axes in the orbital plane (1). The quasi-inertial mode is believed to be an efficient means for maintaining the nominal solar orientation for SPS solar arrays in which attitude error may range only up to 18.8 degrees depending on spacecraft inertia, and thus the cosine loss in insident solar radiation is 5% at most. In the quasi-inertial mode with no presence of energy dissipation, it is shown that the solar incident energy loss induced by gravity gradient torque is small. However, if there is a resistance, such as drag force and/or other frictional force caused by the structural flexibility which is proportional to attitude rate, the motion of the system will approach an equilibrium state. Of course, in this case, the system is no longer conservative. In the latter part of this paper, we have established that none of the equilibrium states are close to the desired attitude state for SPS. In other words, in the presence of energy dissipation the quasi-inertial mode may not be maintained within prescribed tolerance unless a power-assisted control system is used to make up the system energy loss. In the last part of this paper, a fixed terminal time and state optimal control problem is formulated and an algorithm is proposed for the determination of the optimal control which is applied periodically to the satellite to compensate for the attitude loss and phase deviation from the ideal quasi-inertial attitude condition. THE EQUATION OF RIGID BODY MOTION IN SYNCHRONOUS ORBIT Consider a rigid body moving in a circular synchronous orbit. Assume that the only external torque exerted on this body is the gravity-gradient torque. The motion equation is
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