Fig. 7-2. V-Slit Sun Sensor Geometry With Elevation Misalignment

sinwAr

+ tan2e

For small c, we may keep only the first-order terms in c, so that tan(0o+ e) - « costoA/

tanft^o=

sinwA /

Finally, an azimuth misalignment occurs when the two FOV intersections with the spin equator are separated by an angle 8 in the spin plane, as shown in Fig. 7-3. Due to the azimuth misalignment, 8, the actual rotation angle between the two Sun-sighting events is BD rather than BC. Comparing Fig. 7-3 with Figs. 7-1 and 7-2, it is clear that all of the previously derived equations are still valid if «A/ is replaced with «A/ — 8. Thus, from Eq. (7-1), with only the azimuth misalignment we have tan ft =

FOV OF Pr-2 AT ¡UN SIQHTINQ 2

FOV OF Pr-2 AT ¡UN SIQHTINQ 2

FOV OF PF-1 AT SUN StOHTtNO 1

Fig. 7-3. V-Slit Sun Sensor Geometry With Azimuth Misalignment

FOV OF PF-1 AT SUN StOHTtNO 1

Fig. 7-3. V-Slit Sun Sensor Geometry With Azimuth Misalignment

With all possible misalignments, the general expression for the Sun angle can be obtained by replacing 90 with 0O+A0 and <oA/ with <oA/ - 8 in Eq. (7-4). That is, tan ^Aff,«,«*

+ tan2e

For simulation, the inverse expression for Ar as a function of ß and the misalignment angles is cos(wAí-«)=i[6+V¿2-flc ]

¿ = tan t tan(ö0+Aff +1) c = tan2(0o+A0+ «)+ tan2« - tan20

7,1.2 Digital Sensors

As indicated in Section 6.1, one- and two-axis digital sensors are closely related, the former consisting of a command component (A) and a measurement component (£) and the latter consisting of two Gray-coded measurement components (A and B) as shown schematically in Fig. 7-4.

Alignment of Digital Sensors. The alignment of digital sensors consists of two distinct processes. Internal alignment is performed by the sensor manufacturer to ensure that the sensor slits, the Gray-coded reticle patterns, and the alignment mirror form a self-consistent unit. External alignment of the sensor unit relative to the spacecraft attitude reference axes is performed by the spacecraft manufacturer. In this section, we will model only the external alignment and assume that there are no errors in the internal alignment.

The alignment mirror is used to orient the sensor boresight The remaining alignment parameter is the rotation of the sensor about the boresight axis. For single-axis sensors, the command component entrance slit is generally parallel to the spacecraft spin axis. In most cases, two-axis sensors are mounted such that either the A or the B measurement slit is parallel to the spacecraft X- Y plane (see Fig. 7-5).

We define the sensor Z axis, Zs, as the outward normal of the plane containing the alignment mirror and the entrance slits of both components. The

2s-axis is the sensor boresight and is the optical null of . both the A and B components. The Xs and Ys sensor axes are perpendicular to Zs as defined in Fig. 7-4 and Table 7-1. Note that some of the internal alignment parameters could be modeled by treating two-axis sensors as two independently aligned one-axis sensors, although we will not use that model here.

The orientation of a one-axis sensor with boresight located at (4>' = 4>+S4>,S\), misaligned slightly from the nominal location of (<>,0), and rotated through the angle Ify about the boresight, is shown in Fig. 7-5. The transformation which rotates a vector from sensor to spacecraft coordinates may be expressed as the transpose of a 3-2-3 Euler rotation with angles 8,=<j>', 62 = 90° - 6 A, and &3 = Sip,* where 8 k, 8<(>, and &p are small misalignment angles and the 0-deg nominal value of

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