## Rate

attitude state then moves horizontally within the deadband at a constant angular velocity; when the deadband boundary is next crossed, another thrust is commanded. The trajectory, in the absence of environmental torques, for representative initial conditions A, B, and C is shown in Fig. 18-12. The path 7-8-9-10-11,..., is a steady-state trajectory which is approximately maintained by the control laws.

The size of the attitude deadband in state-space is determined by the system gain, K, time constant, t, and minimum thruster duration, At, with the permitted position and rate dimensions given by $0 = 2ht/K and 80 = 2ht/ Kt, respectively, as shown in Fig. 18-12. The mean time, </), between thrusts in steady state depends on the minimum angular rate change and is given by

The general effect of environmental torques is to perturb the attitude, as shown in Fig. 18-12; this results in curved trajectories in 'he state-space diagram.

The HEAO-1 control law has the advantage of simplicity, and gains and deadbands may be selected to suit various applications. The major disadvantages are that (1) the response of the system to disturbance torques is undamped, which results in a waste of expendables as the attitude state is driven within the deadband; and (2) the attitude pointing accuracy is severely limited by the requirement for complex ground-based support to provide periodic updates to the reference attitude, qQ\ typically every 12 hours. One obvious improvement on the HEAO-1 control system is to provide an autonomous capability for updating the reference attitude. Such a system, using a star tracker (see Section 6.4) as the sensing device will be used on HEAO-B [Hoffman, 1976]. In addition to providing periodic reference attitude updates, the HEAO-B control system continuously estimates the gyro drift bias (see Section 7.8) using an onboard version of the Kalman filter discussed in Section 13.S. A similar system for SMM, using a precise Sun sensor as the primary attitude reference, is described by Markley [1978].

### 18 J J Earth-Referenced Spacecraft

The two basic limitations of the HEAO-1 control system—the lack of both damping and an autonomous attitude reference—are easily overcome for Earth-referenced spacecraft. A momentum wheel provides gyroscopic rigidity and thereby permits damping. In addition, the control system may utilize either gravity-gradient torque or horizon sensors to measure absolute position errors and, consequently, does not require extensive ground support.

The spacecraft considered here rotate at one revolution per orbit in an orbit of moderate eccentricity (say, e<0.1) at altitudes where atmospheric drag may be neglected. We will first reformulate Euler's equations for these spacecraft by deriving an expression for the gravity-gradient torque. Next, the general characteristics and approximations underlying the equations are described. Finally, they are applied to GEOS-3, HCMM, SEASAT, and CTS to illustrate the analytical procedures used for the evaluation of arbitrary stabilization systems.

We assume that the nominal mission attitude is as shown in Fig. 2-4, where the body Z axis (normally the payload axis) is along the nadir, and the body Y axis is along the negative orbit normal. For a circular orbit, the body X axis is along the velocity vector.* The attitude angles are defined as roll, pitch, and yaw, which are small rotational errors about the velocity vector, negative orbit normal, and nadir. (Alternatively, these may be thought of as small errors about the body X, Y, and Z axes.) The roll, pitch, and yaw angles (in radians) are denoted by ip, and L. respectively. The transformation matrix from the orbit reference frame to the body frame is,

XB. The order of the three rotations

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