and A4>h and are constants. E is the unit nadir vector.

It remains to express $ and A$5 in terms of their arguments. For the case of the Sun-to-Earth-in model, $■=<!>, and is again calculated from Eq. (7-57) or Eq. (7-59), in which p and y are replaced everywhere by pc + Ap and yN + by, respectively. A$s is calculated by applying Napier's rules for right spherical triangles to the lower triangle associated with the Sun sensor in Fig. 7-16, yielding:

which simplifies to

where ft is computed from

Here, A is the known attitude, and S is determined from an ephemeris and is evaluated at /= ts. Note that this description of horizon sensor biases is valid only for horizon sensors with fixed mounting angles, y.

In the case of panoramic scanners, the nominal mounting angle, yK, varies by fixed increments (Section 6.2) in a plane inclined at an angle tH to its nominal orientation, as shown in Fig. 7-17 for the case of the Earth-in models. Thus, Ay and y, the true mounting angle, are related to the other quantities as follows:

The observation model is again described by Eq. (7-60), in which $ and At are replaced by and At,. A<J>S and <J>W are as before, but now depends on iH (through y) in addition to the other biases, and AOw is defined by

where is a constant azimuth bias on the horizon sensor mounting angle, and A<bHR is an additional horizon sensor rotation angle bias caused by tH and Ay as shown in Fig. 7-17.

is calculated by applying Napier's rules for right spherical triangles to the lower triangle associated with the horizon sensor in Fig. 7-17 to obtain sin [ 90° - (90° -«„)]- tan(A4>w)tan [ 90° - (90° - yN - Ay) ]

which simplifies to sincy,

All of the above applies also to the Sun-to-Earth-out and Sun-to-Earth-midscan models, where t,, and H, are replaced everywhere by <I>0, ta, and H0 or tm, and Hm, as before.

From Fig. 7-17 we see that the geometrical relationship between A/?, and ts is identical with that between A4>„, Ay, and tH. The practical difference is that p can be found directly from Eq. (7-62), whereas y cannot and therefore must be expressed in terms of yN, Ay, and tH. It is the independent knowledge of (3 which makes it possible to eliminate A/? from the expressions for $ and A4>s. Once es is found, A/3 can be computed from /?m, fi, and ts.

The expression on the right side of Eq. (7-60) is a complicated function of the following biases: cs, tH, Ay, Ap, and The values of the various coefficients in that expression depend on the numerical values of the attitude A and the time. To determine the above biases, at least five independent equations are necessary, although the numerical solutions of such a system would generally not be unique. Such equations can be obtained by taking measurements at various attitudes and times. When reasonable initial estimates are available, ambiguities can generally be resolved and satisfactory solutions obtained.

7.4 Modeling Sensor Electronics

7.4.1 Theory

In the previous three sections, ideal mathematical models have been constructed for Sun sensor and horizon sensor systems. Effects of electronics signal processing on the sensor outputs have been considered only in an ad hoc fashion (e.g., the azimuth biases and central body angular radius biases introduced in Section 7.2). In this section we consider the electronics signal processing systems from a more fundamental viewpoint. For a large class of such systems, which we shall assume to include all cases of interest to us, the output signal, Sa(t), is related to the input, 5,(0, by

J-co where h(t,tr) is called the impulse response Junction of the system. A system that obeys Eq. (7-66) is called a linear system, because the response of such a system to the linear input combination Sj = Sn + Sn is the output Sa = S01 + S02, where Sol is related to Sn by Eq. (7-66) and, similarly, for SQ2 and Sl2. Extensive literature exists on the subject of linear systems. See, for example, Schwarz and Friedland [1965], Kaplan [1962], or Hale [1973].

If the input to a linear system is the unit impulse function or the Dirac delta function (see the Preface), s,(0=M'-'o)

then the output is

which is why h is called the impulse response function. A linear system is time invariant if the impulse response depends only on the time elapsed since the application of the input impulse, and not otherwise on the input time. That is,*

For a time-invariant linear system, the input/output relation resulting from combining Eqs. (7-66) and (7-67) is

Any integral of the form r mg(*-sw

is called a convolution integral [Schwarz and Friedland, 1965; Kaplan, 1962; Hale, 1973; Churchill, 1972]. Comparison with Eq. (7-68) shows that the output signal of a linear, time-invariant system is given by the convolution integral of the input signal and the impulse response function. This property will be used shortly.

It is often convenient to work in the frequency domain rather than the time domain. The Fourier transform, X(u), of a function X(t) is defined by

The inverse transformation is given by [Churchill, 1972]

The convolution theorem [Churchill, 1972] states that the Fourier transforms of functions obeying the convolution relation Eq. (7-68) obey the product relation

•This equation means that the function A(/,/") depends only on the difference t-f, and can be written as a function of that single variable. Clearly, the two functions in Eq. (7-67) must be different mathematically, but confusion should not result from using the same symbol for them.

A linear system is causa1 if the output at time t depends only on the input at times f < /; that is, if

All the systems we consider (in particular, that defined by the transfer function of Eq. (7-78)) are causal, and the infinite upper limit of all V integrals can be replaced by /.

The simplicity of Eq. (7-71) as compared with Eq. (7-68) explains the usefulness of analysis in terms of frequency dependence rather than time dependence. Finally, the transfer function is often used in place of h(u) to specify response characteristics of a linear system. 7.4 J Example: IR Horizon Sensor

As an example of the application of sensor electronics modeling, we shall consider the performance of the infrared horizon sensors on the Synchronous Meteorological Satellite-2, SMS-2, launched in February 1975 [Philco-Ford, 1971; Chen and Wertz, 1975]. The input to the sensor electronics system is the intensity of infrared radiation in the 14-16 pm wavelength range falling on the sensor. If we assume that the sensor has uniform sensitivity over its field of view and that the Earth is a uniformly bright disk (a good approximation for this wavelength range, as discussed in Section 4.2), then the input signal is proportional to the overlap area on the celestial sphere between the sensor field of view and the Earth disk.

The sensor field of view is nominally square, 1.1 deg on a side [Philco-Ford, 1971], but for simplicity we model it as a circle with an angular radius of e = 0.62 deg to give the same sensor area. We ignore the oblateness of the Earth (shown in Section 4.3 to be a reasonable approximation) and treat the Earth disk as a circle of radius p= 8.6 deg, the appropriate value for the SMS-2 drift orbit [Chen and Wertz, 1975]. Then the input signal is given by a constant, K, times the overlap area between two small circles on the celestial sphere as shown in Fig. 7-18. Using Eq. (A-14) for the area and the notation of Fig. 7-18, we have

sine sin a

sin t sin p

= Min[2w(l —cosp),2w(l —cose)], a<|p-e| (7.73)

where Min denotes the lesser of the two function values in the brackets. This function is rather intractable mathematically, so we prefer to work with its derivative:

Because the angular radii of the sensor field of view and of the Earth disk, c and p, are constant, the only time dependence in S, is through the time dependence of a(t), the arc-length distance between the centers of the small circles. The horizon sensors on SMS-2 are rigidly mounted on the spacecraft; the motion of their fields of view is due to the spacecraft's spin. Let A in Fig. 7-18 be the spacecraft spin axis, and let the sensor mounting angle and nadir angle be denoted by y and 17, respectively, Then the law of cosines applied to spherical triangle AEO gives cos a ( t ) = cos 7j cos y + sin ij sin y cos $(/) (7-75)

Differentiating Eq. (7-75) gives

1 do = sin n sin y sin 4» d<P sina dt I-cos2« df * '

Substituting Eqs. (7-75) and (7-76) into Eqs. (7-73) and (7-74) gives S, and dS,/dt as functions of the rotation angle, These functions are plotted in Figs. 7-19(a,b) and 7-20(a,b) for ij = 81 deg and ij = 78 deg, respectively, and for y = 86 deg, the mounting angle for the SMS-2 primary Earth sensor [Chen and Wertz, 1975]. The points where the center of the sensor field of view crosses the edge of the Earth disk are indicated by 9, and 00 on the figures.

The calculation of the output signal requires a numerical integration of Eq. (7-68) or its equivalent. Substituting Eq. (7-70) with X = h, and Eq. (7-72) into Eq. (7-70), and then integrating by parts [so we can use Eq. (7-74) rather than Eq. (7-73)] gives*

* A horizon scanner actually makes repeated scans of the Earth, so dS,/df is a rather complicated function. We shall include only one Earth scan in the integral; this is a good approximation if the transfer function is such that the output signal from one Earth scan has decreased to a negligible value before the next scan, as is the case for this example. With this approximation, the integrated part of the integration by parts vanishes at infinite time. The quantity in brackets in Eq. (7-77b) is that integral of the quantity in brackets in Eq. (7-77a) that is finite at a=0.

O.OlVmta|

O.OlVmta|

The spin rate, dO/d/, of SMS-2 was taken to be 600 deg/sec which is close to the measured value [Oien and Wertz, 1973].

The SMS-2 Earth sensor transfer function is [Philco-Ford, 1971]

where Tb= 1.8 ms=detector cutoff r, = 80 ms=preamplifier lower cutoff r2=0.238 ms=preamplifier upper cutoff r3= 2.66 ms=main amplifier lower cutoff r4=0.560 ms=main amplifier upper cutoff T5 = 80 ms = output transformer lower cutoff

For this transfer function, the quantity inside the brackets in Eq. (7-77b) can be evaluated in closed form by the method of residues [Churchill, 1972]. The t' integral in Eq. (7-77b) is then evaluated numerically with dS,/dt given by Eqs. (7-74) through (7-76). The output signals, SQ, for ij = 81 deg and 78 deg are plotted in Fig. 7-19(c) and 7-20(c), respectively. They resemble the curves of dS,/dt to some extent, but the peaks are broadened and time delayed, the positive and negative peaks have unequal height, and SQ does not return to zero between the peaks in the cases where dS,/dt does. Thus, the electronics acts something like a differentiating circuit, although its response is quite a bit more difficult to characterize completely.

The telemetry signal from the SMS-2 horizon sensor is not the output signal, Sa, of the sensor electronics, but rather the time intervals from Sun sightings to Earth-in and -out crossings. The latter times are determined by onboard threshold detection logic [Philco-Ford, 1971]. A negative edge peak detector measures the amplitude of the negative peak of SQ and holds it in the form of a direct-current voltage. The Earth-in crossing is specified as the point where the positive pulse reaches 50 ± 5% of the magnitude of the saved/ peak voltage, and the Earth-out crossing is specified to be where the negative peak voltage is 60 ±5% of the peak. These points are indicated by 4>'; and 4>'c on Figs. 7-19(c) and 7-20(c); and the apparent Earth center, defined as the midpoint between 4>', and 4>'0, is indicated by 4>'£. Note that 4>'£ is displaced from the true Earth center, 4> = 0. Figure 7-21(a) shows % and 4>'0 as a function of nadir angle, tj, plotted at 0.1-deg intervals. This figure also includes a curve showing the rotation angles at which the center of the sensor field of view crosses the Earth's horizon, corrected by a constant offset so that the points fall on this curve for large Earth scan widths (the offset is equal to the value of 4>'£ at large Earth widths). The deviation of the two curves at the left of the figure indicates that modeling iR sensor electronic effects as a fixed bias on the angular radius of the Earth, as discussed in Section 7.2, fails at small Earth widths. Figure 7-21(b) is similar to Fig. 7-21(a) except that it was calculated with 15% and 25% threshold levels for Earth-in and Earth-out times, respectively. The deviations at small Earth widths are exaggerated at these threshold levels, as compared with the nominal levels shown in Fig. 7-21(a). Figure 7-22 shows actual Earth-in and

79 80

NADIR ANGLE (DEG)

Simulated Earth-In and -Out Data as a Function of Nadir Angle, (a) Nominal SMS-2 Triggering Levels; (b) Decreased Triggering Levels. (See text for explanation.)

79 80

NADIR ANGLE (DEG)

Simulated Earth-In and -Out Data as a Function of Nadir Angle, (a) Nominal SMS-2 Triggering Levels; (b) Decreased Triggering Levels. (See text for explanation.)

-out data from SMS-2, which is further described in Section 9.4. The slope of the ellipse is due to orbitai motion effects and is excluded from this section because it is not important for our purposes. What is important is the deviation of the theoretical and experimental points at small Earth widths, called the pagoda effect

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