6 to. 11. 18. 22. ¿6. 30. 38. "J?, lint in nlN'itcs froh SIHHT llne

Fig. 7-22. Actual Earth-In and -Out Data as a Function of Time (SMS-2 Data)

6 to. 11. 18. 22. ¿6. 30. 38. "J?, lint in nlN'itcs froh SIHHT llne

Fig. 7-22. Actual Earth-In and -Out Data as a Function of Time (SMS-2 Data)

because of the characteristic shape of the Earth-out curve [Chen and Wertz, 1975]. The similarity between Figs. 7-21(b) and 7-22 indicates that a more sophisticated treatment of sensor electronics than customarily used in attitude systems may lead to an understanding of the pagoda effect and other related anomalies.

73 Magnetometer Models

This section develops the models used to decode data from fluxgate magnetometers described in Section 6.3 for use in attitude determination algorithms. Equations are included to encode magnetometer data for simulation.

The basic measurement provided by a single-axis fluxgate magnetometer is a voltage, V, related to the component of the local field, H, along the input axis, n, by

where a is the magnetometer scale factor, V0 is the magnetometer bias, and H is thé net local magnetic intensity in body coordinates. The output voltage passes through an analog-to-digital converter for transmission, yielding a discrete output

where Int(x) is the integral part of * and c is the analog-to-digital scale factor.

An alternative system provides a zero-crossing measurement in which a telemetry flag is set when the magnetometer output voltage changes sign. This type of output is used by spinning spacecraft to provide phase information for either attitude determination or control. The flag generally is set and time-tagged in the first telemetry frame following a change in sign.

Vector magnetometer systems consist of three mutually orthogonal, single-axis fluxgate magnetometers. The system can be packaged as a single unit mounted within the spacecraft or attached to a boom, or as separate units dispersed about the spacecraft (for example, on extendable paddles).

The remainder of this section is concerned primarily with vector magnetometers. The models developed are independent of the specific geometry; however, probable biases are highly dependent on both packaging and location within the spacecraft. A magnetometer located at the end of a long boom is unlikely to be exposed to internal magnetic fields but may be misaligned relative to the spacecraft. The opposite is likely to be true for a magnetometer located within the spacecraft interior. In a system consisting of three separate units (particularly units dispersed on extendable hardware) individual units may not be mutually orthogonal and units may be misaligned relative to the spacecraft reference axes.

By analogy with Eq. (7-79), the output of a vector magnetometer system is where the components of V are the outputs of the three units, A is a 3-by-3 matrix, including both scale factor and alignment data, and V0 is the magnetometer bias voltage. Magnetic testing is performed by placing the spacecraft in a Helmholtz coil and measuring the magnetometer response, V, to a systematically varied external field, H. A least-squares fit of the data to Eq. (7-81) yields the 12 parameters, A and V0, which define the magnetometer calibration. The analog output, V, is passed through an analog-to-digital converter to provide the digitized output where r„ cz, and c3 are the analog-to-digital conversion factors.

The matrix A in Eq. (7-81) is diagonal if the three magnetometer input axes are colinear with the spacecraft reference axes and crosstalk is absent. Crosstalk refers to induced magnetic fields normal to an applied field caused by ferromagnetic material or currents in the magnetometer and associated electronics. If crosstalk is absent and the three magnetometer units are mutually orthogonal, then A can be diagonalized by a similarity transformation (see Appendix C). This would imply a coherent misalignment of an orthogonal magnetometer package relative to the spacecraft reference axes. In practice, crosstalk, internal misalignment, and external misalignment cannot be separated and for the remainder of this section we will assume that A is not diagonal.

Int(r, Vf + 0.5) Np= Int(c2P2 + 0.5) Int(c3 V3 + 0.5)

Equation (7-81) may be inverted and combined with Eq. (7-82)_to yield the best estimate of the external magnetic intensity in body coordinates, H,

a2Nn-b2 a3Ny3-b3

where c=(c, + c2+c3)/3 is the mean analog-to-digital scale factor. The components of the matrix B are

Note that Eq. (7-84) is a linear function of the magnetometer output and is thus analogous to the gyroscope model described in Section 7.8. Equation (7-84) assumes that matrix B has a unique inverse (see Appendix Q and requires that no two magnetometer input axes be collinear.

Matrix B defines the effective (not necessarily physical) orientation of the single-axis magnetometers relative to the spacecraft reference axes.* Thus, the effective coelevation, 8„ and azimuth, of the ith magnetometer are

7.5.2 Magnetometer Biases!

Sources of the bias term, V,,, in Eq. (7-83) include magnetic fields generated by spacecraft electronics and electromagnetic torquing coils, and residual magnetic fields caused by, for example, permanent magnets induced in ferromagnetic spacecraft components. It is important to distinguish between the sources of magnetometer bias and of magnetic dipole torque on the spacecraft. Although both are manifestations of uncompensated spacecraft magnetism, a unique relation between the two cannot be derived. The magnetic induction, Bc(x), due to all the localized

'This is a heuristic definition which ignores crosstalk and assumes that misalignment is the source of off-diagonal terms in A.

+ Much of this development follows the formulation of Jackson [1963], where more complete derivations can be found.

current distributions, J(x), contained within the spacecraft can be expressed as the curl of a vector potential, Br(x) = nxA(x), where the vector potential, A(x), is [Jackson, 1963], fio /-J(x')d3x'

Equation (7-90) may be expressed as a multipole expansion using

= £ [ x / 7'(X') dV+^X"/ y-(x')x'dV+--"] (7"92)

For a localized steady-state current distribution, the volume integral of J is zero because n J = 0. Therefore, the first term, which is analogous to the monopole term in electrostatics, is also zero. Manipulation of the lowest order (in 1 /x) nonvanish-ing term in Eq. (7-92) can be shown [Jackson, 1963] to yield llf,

and therefore

Bf(x) = ^ [ 3x(x • m) - m]/x3 + 0 (x "4) (7-93b)

is the magnetic moment of the current distribution J.

The total force on a current distribution, J, in an external field, B, is

and the total torque is

For B constant over the dimensions of the current distribution the net force, F, vanishes and Eq. (7-96) may be reformulated as

The difference between the magnetometer bias and residual spacecraft dipole torque can now be seen from Eqs. (7-93b) and (7-97). A magnetometer bias is a measure of Br{x) in the near field and terms of all order in x contribute because the magnetometer may be in close proximity to magnetic material. However, the residual dipole torque results only from the interaction of the dipole term with the environment because the higher order multipoles do not contribute to the torque.

Magnetometer biases will be induced by magnetic coils used for spacecraft attitude control. These biases may be lessened by winding small coils near the magnetometer in series with the larger control coils to produce a near zero net field independent of the coil current.

To simulate magnetometer biases for prelaunch analysis, or to remove magnetometer biases for postlaunch processing, the field of an electromagnet (see Section 6.7) may be computed as follows. The magnetic induction of a coil with dipole m and radjus a is given by Jackson [1963] as

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