PRECESSION CONTROL Ik » -0.011 J)

-90-60-70-60-50—40-30-20-10 0 10 20 30 40 S0 60 70 60 90 SOUTH LATITUDE NORTH LATITUDE

PRECESSION CONTROL Ik » -0.011 J)

-90-60-70-60-50—40-30-20-10 0 10 20 30 40 S0 60 70 60 90 SOUTH LATITUDE NORTH LATITUDE

Ibl PRECESSION TIME CONSTANT

Fig. 18-17. Control System lime Constants, as a Function of Latitude (Spacecraft is Traveling North). See text for explanation.

For HCMM, the spacecraft parameters satisfy the inequalities

and Eq. (18-90) may be rewritten to good approximation as hua-kp sin2A>0 (I8-92a)

In Fig. 18-17(b), the dotted line shows that the mission attitude (pitch=roll = yaw = 0) is a position of stable equilibrium even in the absence of active precession control, although the time constant is too long (approximately 120 minutes) to counter the effect of typical disturbance torques. The solid line shows the precession time constant for the control law defined by Eq. (18-86). For the HCMM parameters, this precession control law is ineffective in the Southern Hemisphere and unstable between 14° and 76° south latitude when the spacecraft is traveling north. (The Northern Hemisphere is the region of ineffective control system performance when the spacecraft is traveling south.)

Consequently, the HCMM control system deactivates precession control (i.e., sets kp = 0) whenever \BJ Bx\> 1.4. This "magnetic blanking" results in active control only within about 35 deg of the equator. Detailed parametric studies can thus be conducted to establish near-optimal gains and control laws and to obtain regions of stability by solving algebraic equations without the need for time-consuming simulation.

SEASAT. As outlined in Table 18-2, pitch control for SEASAT-A is essentially the same as for HCMM; however, roll/yaw control is achieved using roll and yaw reaction wheels with the control torques based on the horizon scanner roll error signal and wheel speeds [Beach, 1976]:

The commanded roll reaction wheel torque, hx, based on position and rate errors, provides a roll restoring torque, together with damping. The wheel-speed-dependent terms in Eq. (18-93) cancel like terms in Euler's equation and effectively remove the roll and yaw dependence upon the reaction wheel speeds (although, of course, the gyroscopic pitch momentum wheel coupling remains). The yaw wheel is torqued 180-deg out of phase with the roll error signal to provide yaw damping.

Substitution of the wheel control torque, Eq. (18-93) into Eq. (18-53) yields the roll/yaw equation in Laplace transform notation as

with the characteristic equation,

I, I,s< + K>I,s3 + [ 1XK, + I[(KX+K,)+H*y+ (*>*, + W >

where Kx = 4to„2(/- /,) + hua = 0.011 N • m, K, = hua = 0.026 N • m, H = h - I,a0 = 21.3 N m s, and /,= /,= 7 = 25100 kg m2.

The selection of the gains, K„ K-r and K^, is done most conveniently using the root locus plot shown in Fig. 18-18. With all three gains equal to zero, the two roots marked by x are pure imaginary with associated frequencies, 1.5w0 and 3.9w0, related to the nutation and the gravity-gradient torque. As the roll position gain, K„ increases to 0.39 N-m, the roots migrate away from the origin along the imaginary s axis to the point marked by the arrowheads, which implies a faster response to a roll error. Addition of roll rate gain, Kh moves the roots into the third quadrant, which implies damping of both roll and yaw errors. Note, however, that die magnitude of the real part of one root remains small and the associated time constant is therefore large (74 minutes at K,** 116 N-m-s) and, consequently, the yaw damping is slow. The addition of a roll error to yaw torque gain, K^, substantially reduces the longer time constant and thereby improves the system's performance. The gain, 7^=0.08 N-m, is chosen such that the decay constants associated with the two roots are approximately equal, tsb(1.1w0)~ 1 = 14 minutes. The addition of the roll error to the yaw torque gain is fundamental to the WHECON wheel control concept which is frequently encountered in attitude control literature (see. for example, Dougherty, et al., [1968].)

ImlSI

Fig. 18-18. Root LocusPlot For SEASAT Control System. See text for explanation.

Fig. 18-18. Root LocusPlot For SEASAT Control System. See text for explanation.

CTS. The CTS control system, shown in Fig. 18-19, is similar to SEASAT except that gravity-gradient torques are negligible at the synchronous altitude of CTS and thrusters, offset at an angle a from the yaw axis in the roll/yaw plane, are used instead of reaction wheels to control the wheel axis attitude, TTie wheel angular momentum, h, is chosen such that

/»>max[HH(/,-lt )|,k(7, + IX- 7,)|, k(7,-Ix)|,\w0(Ix-7j|] (18-96)

and the roll/yaw equation (Eq. (18-33)) is approximately

Ixs2 + hu„ -hs with the characteristic equation hs

The roots of the characteristic equation are approximately ± ma and ± m„, where un ~ h/iU~< nutation frequency. The control torque, Nr, is based on the roll error signal, derived from horizon sensors,

or in the time domain

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