9-10. Response of Butterworth (Dotted Line) and Least-Squares Quadratic Filters..(^(r)01 + cosur+r, where £(p)=0, e(p)=l/16, u„=0.13 sec-1, and m = 50.)

!' i \ •• •' b \ \ /• ! ' \ \ 1 : \ •/ / V \ I ' \ \

—,-----■ —v—\-------1- —t------V— -------

9-11. Response of Butterworth (Dotted Line) and Least-Squares Quadratic Filters. (K(f)ml + cosui+r, where £(p)=0, c(p) = l/16, um=0.13 sec-1, wc=0.525 sec-1, and m-5.)

93 Scalar Checking

Data validation based on scalar checking occupies an intermediate position in the data validation hierarchy; data must be time tagged and ephemeris information computed, but an attitude estimate is not required. Scalar checking tests the self-consistency of attitude data and is used to remove or correct spurious data prior to the Actual attitude computation.

Scalar checking is based on the elementary principle that scalars, such as the magnitude of a vector or the angle between two vectors, do not depend on the coordinate system in which they are evaluated. In particular, a scalar computed from measurements in the body frame must equal that computed in any convenient reference frame.

The scalar which is validated most frequently in attitude determination systems is the measured'magnitude of the Earth's magnetic field, BM = |BM|.* Although attitude determination algorithms generally require only the measured field direction, the measured magnitude may be compared with the calculated magnitude, UC = |BC|, computed from the spacecraft ephemeris and a model for the Earth's magnetic field (Section 5.1). Measured data is rejected if

where tB is a tolerance parameter based on the magnetometer resolution and unmodeled magnetometer biases.

Comparison of BM and Bc for a data segment is particularly useful for identifying errors in time tagging or in the spacecraft ephemeris. The former error is manifested by a systematic phase difference between Bu(t) and Bc(t) such that Bu(t)ís¿Bc(t + tj, and the latter by a qualitative difference in both amplitude and phase. The root-mean-square of the quantity AB(t¡)= Bu(t,) — Bc(t¡) is a measure of the fidelity of the field model or an indicator of the presence of systematic magnetometer biases. Assuming a magnetometer quantization size of XB, the mean square residual [Coriell, 1975]

must be greater than A^/12. The residual error, 8B=(((AB)2)- X2/I2)l/2, in the IGRF (1968) magnetic field model has been shown to be less than 200 nT (Section 5.1) for intermediate altitudes of 700 to 800 km. Computed residual errors in excess of this value are indicative of unmodeled biases.

For missions which fly magnetometers, Sun sensor data may be validated by comparing the measured angle between the Sun and the magnetic field vectors with

'Rigorously, the measurement BM is a true vector only if the magnetometer triad is orthogonal; otherwise, Bw denotes three ordered measurements which are treated algebraically as a vector.

that computed from the spacecraft ephemeris and a field model. Assuming that BM satisfies Eq. (9-26), Sun data is flagged if

where SM and Sc denote the measured and the calculated Sun vectors and te is a tolerance based on the magnetometer and Sun sensor resolution, the accuracy of the field model, and unmodeled biases. An equation analogous to Eq. (9-27) may be used to obtain a measure of the relative Sun sensor and magnetometer alignment and the error in the model field direction. Expected root-mean-square (rms) residuals, due to Sun sensor and magnetometer resolution, will contribute a residual to Eq. (9-28) analogous to the X\/\2 term in the discussion following Eq. (9-27). However, this residual is highly orbit dependent and can best be established via simulation [Coriell, 1975]. Unmodeled rms residual angular errors in the IGRF reference field are of the order 0.3 to 0.5 degree (Section 5.1) at 700 to 800 km.

Earth horizon scanners and similar devices measure the Earth nadir vector in body coordinates, EM. This vector must satisfy the condition

|cos- '(E„- B„) - cos " '(Ec- Bc)| < tM (9-29)

when used with magnetometer measurements, and

when used with Sun sensor measurements. Ec is the nadir vector in inertial coordinates and cM and es are tolerances associated with the magnetometer and Sun sensor accuracies, respectively.

Clearly, mean and root-mean-square residuals of scalar quantities are useful for assessing the magnitude of unmodeled errors in sensor data. Displays of predicted-versus-observed scalars are useful in identifying time-tagging or other systematic errors in the data, particularly before mission mode when tests based on an a priori attitude are not available.

In addition to its use in validating and assessing sensor data, scalar checking has been used in star identification (Section 7.7), magnetometer bias estimation [Gambhir, 1975], and Sun sensor data reduction. For example, a procedure to periodically compensate for magnetometer biases is based on the assumption that the biases are constant for some appropriate time interval (typically an hour or more). Neglecting noise, we may write

where B(,((,) is the unbiased measured magnetic field at time /,, Bis the biased measured magnetic field at time /„ and b is the magnetometer bias which is assumed constant in time. Although the components of the vector Bv(ti) are attitude dependent, the magnitude B0(Q is not. Assuming that the magnetometer triad is orthogonal, we may equate B0(tj) to the model field magnitude, Bc(t,),

Bh ( 0 = Bl (/,) = |B„( *,) - b|2 = + ¿>2 - 2b - B„ (9-32)

where the explicit time dependence of Bc and BM has been suppressed for convenience.

The vectors BM are known from measurements, and the corresponding values of Bc can be calculated using spacecraft ephemerides and geomagnetic field models. Therefore, the values of Y(t,) corresponding to each value of BM can be calculated, and a least-squares fit of the data to Eq. (9-33) (see Chapter 13 and Gambhir, [1975]) can be made to obtain the best estimates of the three components of b.

As a second application, a scalar test may be applied to correct anomalous two-axis Sun sensor data. Figure 9-12 illustrates a problem encountered with the

REC • * |

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