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It follows that the vector <ogM, representing the collective output of the gyro configuration, is given by
7.83 Calculation of Estimated Angular Velocity From the Gyro Measurements
An expression for the true spacecraft angular velocity is found by solving Eq. (7138) for <o,
where
C=[UrKrKU] XUrK
In the case of three gyros, this expression reduces to c=[Kuyx
Because the expected value of ng is zero, an estimate of the spacecraft angular velocity, <«>, becomes where b= Cb^ is the effective gyro drift rate vector in the spacecraft frame.
In practice, the scale factor corrections and input axis orientations are not known precisely and will vary slightly with time. The matrix C is calculated before launch based upon ground calibrated values for K and U and then remains invariant. To take time variations into account, a misalignment/scale factor correction matrix, G, is introduced such that where input estimates for b and G are used. Equation (7141) is a convenient algorithm for. the calculation of spacecraft angular velocities from gyro measurements for use in attitude propagation (Section 17.1) or in attitude control (Section 19.4).
Because each gyro contributes a scale factor uncertainty and a 2degoffreedom input axis alignment error, the elements of G will in general be independent for N >3. G may be initialized at zero if C contains all scale factor and alignment information after gyro calibration and spacecraft assembly. After launch, it may be necessary to refine G occasionally dae to small scale factor and alignment changes. Estimates for the gyro drift are available after gyro calibration but must be redetermined frequently after launch. Procedures for the refinement of b and G are presented in Section 13.4 and by Gray, el al., [1976].
7.8.4 Modeling Gyro Noise Effects
Gyro noise may seriously degrade the accuracy of the calculated spacecraft angular velocities and of attitude estimates based on these angular velocities. For torque rebalanced gyros, it is convenient to model gyro noise as being composed of electronic noise, float torque noise, and float torque derivative noise, as introduced in Section 6.5. The models given here for these three noise sources follow the formulation of Farrenkopf [1974] and McElroy [1975] for rate integrating gyros in the rate mode.
Electronic noise is modeled as a timecorrelated colored noise* of standard deviation ae on the gyro output. At the fcth readout time interval, the electronic noise, ne(k), is ne(k) = e«>"[n.(k 1) o.^k I)] + (1 y/le'^ )oeie(k) (7142)
where ?,,(£) is a normally distributed random number with zero mean and unit standard deviation, t is the torque rebalance loop time constant (Section 6.5), and St, is the gyro readout time interval. If t is much less than St,, then Eq. (7142) is simply
Float torque noise is assumed to be white Gaussian noise of standard deviation av on the gyro drift rate. It is modeled as a noise, nc(k), on the gyro output corresponding to the &th readout interval given by nv(k) = aJsiiSv(k)
where $v(k) is a normally distributed random number with zero mean and unit standard deviation independent from Se(k).
Float torque derivative noise is integrated white noise of standard deviation a„, and is modeled as a noise, nu(k), on the drift rate at the fcth readout interval. Thus,
where $u(k) is a random number analogous to but independent of both L(k) and
Gyro noise effects cause an uncertainty in the angular rates calculated from Eqs. (7132) and (7141), which then cause cumulative uncertainties in attitudes determined using these angular rates. If the spacecraft attitude and drift rate are known exactly at time /„ then at time /2 = /,+A/ the attitude uncertainty will follow a Gaussian probability distribution with standard deviation
As an example of noise levels, the HEAO1 gyros have specified values of ae=0.5 arcsec, a„=0.22 arcsec/secl/2, and o„=4.7xl0~5 arcsec/sec372. A plot of Eq. (7143) using these parameters is shown in Fig. 727. At time (0.32 sec) on this figure, attitude uncertainty at the 1 a level is 0.5 arcsec; at t2 (32 min) it is 9.6 arcsec; and at t3 (24 hours), 690 arcsec. The frequency at which the attitude reference and drift rate must be redetermined depends largely on the noise characteristics of the particular gyros in use.
* If the value of a noise at one time influences its value at some other time, then it is a colored noise. The value of while noise at one time gives no information regarding its value at any other time.
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