19.1 Spin Axis Magnetic Coil Maneuvers

19.2 Spin Plane Magnetic Coil Maneuvers

Momentum and Attitude Maneuvers, Optima! Command Procedures, Representative Example of AE-5 Magnetic Maneuvers

19J Gas Jet Maneuvers

19.4 Inertial Guidance Maneuvers Single-Axis Stem, Multiple-Axis Slews

19.5 Attitude Acquisition

Classification of Attitude Acquisition, Acquisition Maneuvers, Representative Acquisition Sequence

This chapter describes procedures for reorienting a spacecraft from one attitude to another. Sections 19.1 and 19.2 describe maneuvers using magnetic coils and Section 19.3 describes maneuvers using gas jets. Section 19.4 then describes procedures for inertial guidance maneuvers. Finally, Section 19.5 discusses the special class of attitude acquisition maneuvers in which the spacecraft starts in an unknown or uncontrolled attitude and ends in an attitude appropriate for mission operations. This chapter uses the general attitude control concepts introduced in Section 15.3 as well as the equations of spacecraft motion presented in Sections 16.1 and 16.2.

19.1 Spin Axis Magnetic Coil Maneuvers

In this section we consider precessional motion generated by a magnetic coil wound around the spin axis of a nonnutating spin-stabilized spacecraft. For such a spacecraft, the angular momentum L can be expressed as

where L is the magnitude of the angular momentum and s=<o is a unit vector along the spin axis. The magnetic moment, M, of an electromagnet aligned with the spin axis (i.e., the spin-axis-coil) may be expressed as

where m0 is the maximum attainable magnetic moment, and u is a commandable coil state parameter which is proportional to the current through the coil and is either positive or negative depending on whether the direction of current flow is counterclockwise or clockwise relative to s (see Section 6.7). The magnetic dipole generated by tiie coil interacts with the geomagnetic field, B, to produce a torque, N, on the spacecraft, given by (see, for example, Jackson [1965])

By definition, the time rate of change of angular momentum is equal to the total impressed torque, i.e.,

From Eq. (19-3), N is orthogonal to both B and M. Because M is either parallel or antiparallel to s, the torque is also orthogonal to L. Therefore, the magnitude of L remains constant,, so that

Combining Eqs. (19-3) through (19-5) gives HS / mnu \

where

This is a well-known equation (see, for example, Goldstein, [1950]), describing the precession of s about the magnetic field, B, with an angular velocity, Qp, which is either parallel or antiparallel to B depending on the sign of the coil state parameter, u. This is illustrated in Fig. 19-1.

It is instructive to express Eq. (19-6) in terms of time rates of change of the right ascension, a, and declination, S, of the spin axis. In terms of these quantities, the celestial rectangular components of s and ds/d/ can be written as s = cos 6 cos ax + cos 8 sin a J + sin Si ds sda. dS. C9"7)

where x, is a unit vector along zXs and is given by

Xj = - sin ax + cos ay Similarly, y, is a unit vector along sXx, and is given by ys=s X ks = — sin S cos ax — sin S sin ay + cos Sz Taking components of Eq. (19-6) along xs and y,, we obtain da urn.

Here, Bx, By, Bt are the celestial rectangular components of the geomagnetic field.

Figure 19-2 provides a physical interpretation of Eqs. (19-8) and (19-9). The torque component along x, rotates the equatorial projection of the spin vector around the celestial z axis and therefore is the cause of right ascension change. Similarly, the torque component along ys pulls the spin axis toward the celestial z axis, which results in declination change.

Integration of Eqs. (19-8) and (19-9) to accurately predict the total spin axis motion requires an accurate knowledge of the geomagnetic field. However, the analytical characteristics of magnetic control maneuvers may be obtained from the dipole model presented in Appendix H. We discuss spin axis maneuvers for two limiting cases of satellite orbits: equatorial and polar.

PROJECTION OF SPIN AXIS INTO EARTH'S EQUATORIAL PLANE

PROJECTION OF SPIN AXIS INTO EARTH'S EQUATORIAL PLANE

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