The shape of the transient response of the system is another design criterion and is controlled by the damping ratio, p. The gain factor, K, required to give a specified damping ratio, is calculated from the root locus diagram by drawing a line from the origin at an angle of ± 0 with the negative real axis, where 0 = arc cos (p). The gain at the point of intersection of this line with the root locus is the required value of K.

For the example shown in Fig. 18-7, the K^ is calculated for a pole located near s = ± i; that is,

The gain required for a damping factor of p = 0.3 is obtained by drawing a line at an angle 9 = arc cos (0.3) = 72.5 deg to the real axis as shown in Fig. 18-7. When the p = 0.3 line is drawn, it intersects the root locus for K >0 at a point s3 near where the lines j»==(jc+3)tan 60° and y= — x tan 72.5° intersect. The solution for this intersection is x = Re(s3) = - 1.06 j»=Im(j3)=3.36

By inspection of Fig. 18-7, the p=0.3 line cannot intersect the locus for K<0 on the left-hand side of the $-plane so that the system is stable with a damping ratio p = 0.3 and K= +60.5.

18.2 Momentum and Reaction Wheels

As discussed in Sections 6.6 and 15.3, momentum and reaction wheels are used to provide attitude stability and control. Various wheel arrangements are,, used. For example, the momentum bias control system includes one or moie momentum wheels to provide a bias, or nominal angular momentum different from zero. This design is often used on Earth-oriented spacecraft, such as the ITOS and AE series, to provide continuous scanning over the Earth. This design is sometimes called a dual-spin spacecraft to indicate that it has two parts rotating at different rates. One component may be completely despun, or rotating at a controlled rate, such as one revolution per orbit such that it maintains the same side pointing toward the Earth.

OAO and IUE are examples of an alternative arrangement in which a system of three orthogonal reaction wheels, with control signals from a set of gyroscopes, is used to provide three-axis stability and high pointing accuracy. This type of system can operate completely despun, with the reaction wheels absorbing all disturbance torques. It can also serve to reorient the spacecraft to a new target attitude by performing a series of slew maneuvers, or rotations about a reaction wheel axis. A hybrid configuration, flown on the Nimbus series, consists of a pitch momentum wheel with reaction wheels in the roll-yaw plane to absorb cyclic torques.

In a momentum bias control system, a momentum wheel is spun up to maintain a large angular momentum relative to disturbance torques. This design is common in Earth-oriented spacecraft where the momentum wheel is along the pitch axis, nominally parallel to orbit normal. The advantages of the momentum bias design are: (1) short-term stability against disturbance torques, similar to spin stabilization; (2) roll-yaw coupling that permits yaw angle stabilization without a yaw sensor for pitch axis pointing; (3) a momentum wheel that may be used as an actuator for pitch angle control; and (4) a momentum wheel that may be used to provide scanning motion across the celestial sphere for a horizon sensor. Thus, momentum bias systems can provide three-axis control with less instrumentation than a three-axis reaction wheel system.

By incorporating horizon scanners into the momentum wheel as described in Section 6.2, roll and pitch euor signals may be provided to the control system as on the ITOS and AE series. Yaw control can be achieved without a yaw sensor through the kinematics of quarter-orbit gyroscopic coupling as shown in Fig. 18-8.

Fig. 18-8. Interchange of Yaw and Roll Attitude Components for a Momentum Wheel With Angular Momentum, h. Fixed in Inertial Space. The yaw error when the spacecraft is at A becomes a roll error when the spacecraft moves to B. (Compare with Fig. 2-4.)

Fig. 18-8. Interchange of Yaw and Roll Attitude Components for a Momentum Wheel With Angular Momentum, h. Fixed in Inertial Space. The yaw error when the spacecraft is at A becomes a roll error when the spacecraft moves to B. (Compare with Fig. 2-4.)

Here, a yaw error, at one point in the orbit becomes a roll error, a quarter of an orbit later.

In a typical momentum bias system, closed-loop pitch angle control is maintained by comparing a pitch index fixed in the spacecraft body to the midscan horizon-crossing signal (see Section 6.2). Open-loop roll control is often performed using magnetic coils, as on AE or ITOS. In the AE system, the attitude is determined on the ground, and magnetic coil commands are generated to null the roll error by reorienting the pitch axis toward orbit normal. In addition to compensating for attitude disturbance torques, adjustment must also be made for the change in direction of the orbit normal due to precession of the orbit (see Section 3.4). Transferring momentum between the wheel and the spacecraft body to change the body spin rate may be used for switching between spining and nonspinning operations or for changing the pitch angle in the despun mode. The Component of the total angular momentum, LP, about the pitch axis is given by

where Ip is the moment of inertia of the body of the spacecraft about the pitch axis, wp is the body spin rate about the pitch axis, and h is the angular momentum of the pitch wheel where the wheel momentum is oriented along the positive pitch axis. From conservation of angular momentum, the change in body rate due to a change in wheel momentum is a AA

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