Page 220 - Modern Control Systems

Chapter 3 State Variable Models

scientific instruments pointing up will see deep space, as desired. To achieve earth-
pointing attitude, the spacecraft needs an attitude hold control system capable of
applying the necessary torques. The torques are the inputs to the system, in this case,
the space station. The attitude is the output of the system. The International Space
Station employs control moment gyros and reaction control jets as actuators to con-
trol the attitude. The control moment gyros are momentum exchangers and are
preferable to reaction control jets because they do not expend fuel. They are actua-
tors that consist of a constant-rate flywheel mounted on a set of gimbals. The fly-
wheel orientation is varied by rotating the gimbals, resulting in a change in direction
of the flywheel angular momentum. In accord with the basic principle of conserva-
tion of angular momentum, changes in control moment gyro momentum must be
transferred to the space station, thereby producing a reaction torque. The reaction
torque can be employed to control the space station attitude. However, there is a
maximum limit of control that can be provided by the control moment gyro. When
that maximum is attained, the device is said to have reached saturation. So, while
control moment gyros do not expend fuel, they can provide only a limited amount
of control. In practice, it is possible to control the attitude of the space station while
simultaneously desaturating the control moment gyros.
Several methods for desaturating the control moment gyros are available, but
using existing natural environmental torques is the preferred method because it mini-
mizes the use of the reaction control jets. A clever idea is to use gravity gradient
torques (which occur naturally and come free of charge) to continuously desaturate
the momentum exchange devices. Due to the variation of the earth's gravitational
field over the International Space Station, the total moment generated by the gravita-
tional forces about the spacecraft's center of mass is nonzero. This nonzero moment is
called the gravity gradient torque. A change in attitude changes the gravity gradient
torque acting on the vehicle. Thus, combining attitude control and momentum man-
agement becomes a matter of compromise.
The elements of the design process emphasized in this example are illustrated in
Figure 3.28. We can begin the modeling process by defining the attitude of the space
station using the three angles, 0 2 (the pitch angle), 0 3 (the yaw angle), and di (the roll
angle). These three angles represent the attitude of the space station relative to the
desired earth-pointing attitude. When &i = 6 2 = d 3 = 0, the space station is oriented
in the desired direction. The goal is to keep the space station oriented in the desired
attitude while minimizing the amount of momentum exchange required by the con-
trol momentum gyros (keeping in mind that we want to avoid saturation). The con-
trol goal can be stated as

Control Goal
Minimize the roll, yaw, and pitch angles in the presence of persistent external dis-
turbances while simultaneously minimizing the control moment gyro momentum.
The time rate of change of the angular momentum of a body about its center of
mass is equal to the sum of the external torques acting on that body. Thus the atti-
tude dynamics of a spacecraft are driven by externally acting torques. The main
external torque acting on the space station is due to gravity. Since we treat the
earth as a point mass, the gravity gradient torque [30] acting on the spacecraft is
given by