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^CORRESPONDS TO 32 OSQLLATIDN PERIODS OVER THE »32-DEG FIELD OF VIEW f f CORRESPONDS TO 64 OSCILLATION PERIODS OVER THE >32—DEG FIELD OF VIEW.

^CORRESPONDS TO 32 OSQLLATIDN PERIODS OVER THE »32-DEG FIELD OF VIEW f f CORRESPONDS TO 64 OSCILLATION PERIODS OVER THE >32—DEG FIELD OF VIEW.

7 2 Horizon Sensor Models

### Steven G. Hotovy

In this section, we provide several observation models for any sensor which scans the celestial sphere in a small circle and is sensitive to the presence of electromagnetic radiation from a body in its field of view. Such sensors, described in Section 6.2, may be divided into three categories:

1. Body-Mounted Sensor (BHS)—a visual or infrared telescope fixed on the body of a spinning spacecraft

2. Panoramic Scanner (PS)—a visual scanner operating in the scan mode on a despun spacecraft

3. Wheel-Mounted Sensor (WHS)—an infrared scanner consisting of a bolometer attached to the body of the spacecraft into which the field of view of a lens or mirror mounted on a rapidly spinning wheel is reflected

### 7.2.1 Horizon Sensor Geometry

Figure 7-11 depicts the movement of the optical axis of a BHS sensor as the spacecraft spins. A is the spin axis attitude, X is a reference point in the spacecraft body, $>p is the azimuth, and yN is the nominal coelevation of the sensor optical axis in body coordinates. As the optical axis, P, sweeps through the sky, the sensor detects an in-crossing (entrance of the central body into the sensor field of view) at time t, at the point H, = P(/,). At some later time, t0, it will detect an out-crossing (departure of the central body from the sensor field of view) at the point H0

The same geometry applies to a wheel-mounted scanner, with some change in interpretation. In this case, A is the spin axis of the wheel and X is some reference vector in the spacecraft body which lies in the plane of the wheel. As the mirror rotates, the sensor detects an in-crossing and an out-crossing as before. However, for a wheel-mounted sensor, a magnetic pickup is mounted on the body.of the spacecraft at some index point and a magnet is mounted on the wheel. These are used to measure the time of one complete revolution of the wheel. As a result, a wheel-mounted sensor can measure the central body width, equal to the rotation angle about A from H7 to H0. In addition, it can measure the split-to-index time, tsl, the time between the detection of the midscan of the central object, 0.5(/o+/,), and the detection of the magnet by the magnetic pickup, t^^; that is

1.2J. Nadir Vector Projection Model for Body-Mounted Sensor

The nadir vector projection model for a BHS is

where P is the unit vector along the line of sight of the sensor, E is the nadir vector to the central body, and p is the apparent angular radius of the central body as seen from the satellite. The value of E-P will oscillate sinusoidally approximately once per spacecraft rotation as P sweeps through the sky. The value of E-P —cosp will be zero when the angle between E and P is equal to the_ apparent _ angular radius of the central body, i.e., at horizon crossing times when P = H, or H0.

The values of E and p may be determined from an ephemeris, which provides the spacecraft-to-central body vector, E. If we assume that the central body is a sphere, then p satisfies cos p = (E2-R¿)W2/E

where RE is the radius of the central body. If the central body is the Earth, oblateness may be considered (procedures for modeling an oblate central body are discussed in Section 4.3), in which case RE in Eq. (7-40) is latitude dependent.

To evaluate Eq. (7-39), it is necessary to express P in inertial coordinates. This expression is given in terms of the sensor location in the spacecraft frame a.id the spacecraft orientation in inertial space. For this model, we assume that the spacecraft has an inertially fixed spin axis and is spinning at a constant rate (i.e., nutation, coning, precession, and spin rate variations are assumed to be negligible). The pertinent parameters are the initial phase of the spacecraft 4>0 at time r0; the spin rate, and the spin axis vector, A. The phase at a time t is, then,

<D, = <1>0+ «(/-/„) The position of the center line of sight in spacecraft coordinates is

cosy where y = yN + Ay is the true mounting angle.

The attitude matrix, £(/), for a spinning spacecraft at time t is given by (see

Section 12.2 and Appendix E)

A ,A 3cos —A 2sin <£, A2A3cos <b, + A ,sin

- A ¡A 3sin <&,-A 2cos - A 2A 3sin Q, + A ,cos

where \=(AvAl,A3jT, the spin axis unit vector, is now expressed in inertial coordinates and is as in Eq. (7-41). Thus, the location of the line of sight of the sensor at time t in inertial coordinates is

This is then substituted into Eq. (7-39). This model is not valid in the case of terminator crossings for a visible light sensor; thus, terminator rejection is required.

7.23 Central Body Width Model

In the case of valid in- and out-crossings from a BHS or a PS, we may develop a model incorporating both crossing times t, and ta. This model is

where u is the body rate (again assumed to be constant), n is the number of complete spacecraft rotations between t, and tQ, and fl is the central body width (in degrees), which can be calculated as follows.

Applying the law of cosines to spherical triangle A EH in Fig. 7-11, we obtain cosp = cos y C0S7J + sin y sinij cos^ y j (7-46)

which becomes, upon solving for 52,

Here y = yAi + Ay, where Ay is a fixed mounting angle bias. A fixed bias can similarly be included in p.

When other effects (such as oblateness or height of the C02 layer) are considered, the expression for S2 becomes

where p, and pQ are the effective scan-in and scan-out radii of the central body, including all correction factors to the nominal radius.

For a WHS, the central body width can often be obtained directly from telemetry data. The scanners aboard SMS-1 and -2 and AE-3, -4, and -5, for example, provided the Earth-in and -out times, t, and t0, and the wheel speed, From Eq. (7-45), we have

On other spacecraft (SAS-3, for example), the telemetry data consisted of a voltage which was converted to an Earth width, G, from a calibration curve.

The assumptions and limitations for the nadir vector projection model hold true for the central body width model as well. In addition, we must assume that the orbital motion of the spacecraft is negligible between in- and out-crossings. This effect is more troublesome for BHSs and PSs than for WHSs because wheel rates are generally much faster than spacecraft body rates.

Knowledge of 0 permits the calculation of the nadir angle, r\. Equation (7-46) leads to a quadratic equation in cos 7/ with solutions cos y cos p ± Ar(cos*y + A:2-cos2p),/2

Because both solutions are geometrically meaningful, more information, such as an a priori attitude estimate, is needed to resolve the ambiguity. Once it has been relsolved, however, we know that the spin axis of the spacecraft (or of the wheel in the case of a WHS) lies on the cone in inertial space centered on E and of radius ij.

### 12.4 Split Angle Model for Wheel-Horizon Scanner

As mentioned previously, a wheel-mounted scanner provides two readings that are not available from a body-mounted sensor: the wheel rate, uw, and the split-to-index time, tsl. These can be combined to determine the azimuth, a, of the magnetic pick-off relative to the midscan of the central body. As shown in Fig. 7-12, we have a = uwtsl + Aa (7-51)

where Aa is the azimuthal misalignment of the pickoff from its nominal value. This can be combined with the spin axis attitude to determine the three-axis attitude of the spacecraft, since the spin angle model specifies the azimuthal orientation of the spacecraft body about the wheel spin axis.

PTCKOFF (TRUE)

PICKOFF {NOMINAL!

PTCKOFF (TRUE)

PICKOFF {NOMINAL!

72S Biases

The model developed above may not accurately explain sensor behavior because of the presence of additional sensor bias».* For example, there may be an azimuthal mounting angle bias, AO, due to either a mounting misalignment or incorrectly calibrated sensor electronics. (See Section 7.4.) This bias can be added to the nadir vector projection model by replacing with 4^+AG» in Eqs. (7-42) and (7-44). If this bias is due to sensoT electronics, it may be appropriate to use separate in- and out-crossing biases, A$y and A$0, since the electronic response may be different in these two cases. This may be incorporated into the central body-width model by changing Eq. (7-45) to

Another possible bias is a systematic variation, Ap, in the angular radius of the central body. This may be caused by a genuine uncertainty in the size of the effective triggering radius of the central body itself, or, more likely, may reflect the sensor triggering performance as shown in Fig. 7-13. Under nominal circumstances, we assume that the FOV of the sensor is circular and that the sensor will register an

Fig. 7-13. Bias on Angular Radius of the Central Body in- or out-crossing when the central body occupies 50 percent of the FOV. However, if the sensor triggers at some value other than 50 percent, the effective size of the central body changes. In Fig. 7-13, the horizon sensor triggers when the central body occupies only about 10 percent of the FOV. This means that the apparent size of the central body is greater than the actual size. Note that Ap is independent of the path of the sensor across the central body, although the

* Each of the biases described here has been found to have a significant effect on real data for some missions.

Fig. 7-13. Bias on Angular Radius of the Central Body difference in triggering times will vary with the path. This effect can be added to the nadir vector projection and central body width models by replacing p with p+Ap in Eqs. (7-39), (7-46) through (7-48), and (7-50).

Finally, for a WHS, the optical axis of the bolometer (see Section 6.2) mounted on the body may be misaligned relative to the spin axis of the wheel. This results in a sinusoidal oscillation of the central body width data with a frequency equal to the body spin rate relative to the central body. This phenomenon was first observed on the AE-3 spacecraft [Wertz, ei a!., 1975]. The phase and amplitude of the oscillation will depend on the phase and amplitude of the bolometer misalignment, as shown in Fig. 7-14. Here, S is the spin axis of the wheel; Bi and B2 are the positions of the bolometer optical axis at times and f2; H, and H0 are the in-and out-crossings of the bolometer at time /,; and M, and M0 are the positions of the mirror normal at these times. Figure 7-14 shows that the bolometer 2 Earth width, which is the rotation angle about the spin axis from M, to M0j, is greater than that from bolometer 1. The nadir angle/Earth width model, Eq. (7-50), can be changed to reflect a bolometer offset, although the derivation of this new model is not straightforward [Wertz, et a/., 1975; Liu and Wertz, 1974], The model is cos p =■ cos a (cos y cos ij + sin y sin ij cos L,)

+ sino{[sinycosij + (l-cosTr)sinTjcosL/]cos(fi- L/)-sinijcosfi } (7-53a)

+ sine {[siny cost/+ (1 — cosy )si n 7j cosL0]cos(B+ L0)-sintjcosi? } (7-53b)

where o is the offset angle between the bolometer and the spin axes, B is the rotation angle about the spin axis from the bolometer axis to the nadir vector, L, is

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