• OFFSET ANGLE BETWEEN ANGULAR MOMENTUM VECTOR AND ORBIT NORMAL.

1 NUTATION HALFTONE ANGLE. T>> - ARCTAN ([<lx"xl2 ♦ «2"Z)J] '"/'^„Sl), WHERE lx. Iz. AND ARE THE X. Z. AND WHEEL INERTIAS: u>x AND Uy THE BODY RATES IDEG/S): AND S. THE WHEEL SPEED IRPMI.

• OFFSET ANGLE BETWEEN ANGULAR MOMENTUM VECTOR AND ORBIT NORMAL.

1 NUTATION HALFTONE ANGLE. T>> - ARCTAN ([<lx"xl2 ♦ «2"Z)J] '"/'^„Sl), WHERE lx. Iz. AND ARE THE X. Z. AND WHEEL INERTIAS: u>x AND Uy THE BODY RATES IDEG/S): AND S. THE WHEEL SPEED IRPMI.

deadbeat, meaning no recoil, after the stroke employed by drummers, and are based on the conservation of angular momentum. Consider a spacecraft librating under the influence of gravity-gradient torques about the pitch axis with a boom fully extended along the yaw axis. At any time, the attitude state may be represented as a point in the pitch/pitch rate state-space as shown in Fig. 19-16. If an initially extended boom is retracted to an intermediate length at near zero pitch and minimum pitch rate, the decrease in inertia about the pitch axis will cause the pitch rate to increase (become less negative to conserve angular momentum) and follow the trajectory depicted by the inner circle.* If subsequently the boom is reextended at a pitch angle near zero and maximum pitch rate, the increase in inertia will reduce the pitch rate and remove the pitch librations. The proper choice of an intermediate moment of inertia, /„ is derived as follows. Assume that the retraction and extension maneuvers are instantaneous. Conservation of angular momentum at retraction requires

Fig. 19-16. Deadbeat Maneuver tor Removal of Pitch Libration Using Extendable Boom. The origin of the figure corresponds to pitch »pitch rate 0, but the scale of the axes is arbitrary. Note that pitch=0 implies an inertial rate about the pitch axis of minus 1 revolution pet orbit (rpo) or 1 rpo almut the positive orbit normal.

Fig. 19-16. Deadbeat Maneuver tor Removal of Pitch Libration Using Extendable Boom. The origin of the figure corresponds to pitch »pitch rate 0, but the scale of the axes is arbitrary. Note that pitch=0 implies an inertial rate about the pitch axis of minus 1 revolution pet orbit (rpo) or 1 rpo almut the positive orbit normal.

•We assume that the inertia change is instantaneous. For typical configurations, boom maneuvers require I to 10 minutes whereas libration periods are typically 1 hour («orbital period/V3 ). The external torques are proportional to pitch (and near zero) when the extension and retraction occur.

where Ie is the moment of inertia about the pitch axis with the boom extended, pt and x are the pitch rate amplitudes before and after extension, and u„ is the orbital angular velocity. At extension, for the pitch rate to vanish, conservation of angular momentum yields

Substitution of Eq. (19-91) into Eq. (19-90) yields

which is the required moment of inertia after retraction to remove a libration rate amplitude />,.

The equation of motion for pitch, p, considering only gravity-gradient torque and assuming a small roll and roll rate, is (see Section 18.3)

where Iy, lx, and lt are the moments of inertia about the body pitch (y), roll (jc), and yaw (z) axes and ua is the orbital rate. Letting T=upt where

Equation (19-93) can be rewritten as

with the integral

where A, the maximum value of pitch, is an integration constant.

For a pencil-shaped spacecraft, Ixa£ly~»It, and w^«^«„. The maximum rate occurs when p = 0, so that

and Eq. (19-92) may be rewritten in terms of the libration amplitude as

Magnetic Stabilization. As a final acquisition maneuver type, magnetic stabilization is a technique in which a spacecraft axis is induced to track the Earth's magnetic field about the orbit. This is used for high inclination spacecraft to provide a reference angular momentum direction normal to the orbit plane. Consider a spacecraft in a polar orbit with an electromagnet along the yaw axis and an onboard damper. Regardless of the initial attitude and attitude rate, the interaction of the electromagnet and external field will cause the yaw axis to track the field (minimum energy configuration) and induce an average spin rate of 2 rpo about the orbit normal. Magnetic stabilization was used for GEOS-2 and proposed for GEOS-3 as the first step in the attitude acquisition because it is passive (no ground support is required) and converts a random initial state into a well-defined state suitable for subsequent acquisition maneuvers.

In this section, we describe the attitude acquisition sequence employed for the Geodynamics Experimental Ocean Satellite, GEOS-3, launched on April 9, 1975, from Vandenberg Air Force Base, California, on a Delta 1410 rocket. Other acquisition sequences are described by Basset [1976] for CTS, Byrne, et al., [1978] for HCMM, and Markley [1978] for SMM. GEOS-3 demonstrated the utility of spaceborne radar altimeters for oceanography and served as a bridge between the earlier geodetic satellites, GEOS-I and GEOS-2, and the ocean resources program, SEASAT. The spacecraft, illustrated in Fig. 19-17, was placed in a circular orbit at an altitude of 843 km and an inclination of 115 deg to provide coverage of the North Atlantic Ocean, the area of primary experimental interest.

The ground-based, open loop attitude acquisition sequence Tor GEOS-3 was designed to achieve a gravity-gradient stabilized, three-axis attitude with the spacecraft z and y axes in the nadir and negative orbit normal directions, respectively. The GEOS-3 control hardware consisted of a 6.5-m boom extendable along the negative z axis; a passive, magnetically anchored eddy current damper (Section 18.4) located at the end of the boom; a z axis electromagnet; and a momentum wheel with its axis along the y axis. Attitude determination hardware consisted of two-axis digital Sun sensors and magnetometers. Pitch and roll stability in the mission mode was provided by gravity-gradient torque and yaw stability was accomplished via quarter-orbit coupling with roll through the momentum wheel (see Section 18.2).

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