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Procedure for Finding the Horizon Crossing Vector. A frequent calculation required for attitude analysis is the determination of the horizon crossing vector, H, as seen by a sensor with a conical field of view. Any point on the horizon lies at the intersection of the horizon and Earth spheroids. To find the two particular horizon points where a sensor first and last senses the Earth, a third surface is needed. If the Sun angle and the Sun-to-Earth rotation angle are available (Eq. (7-57)), that surface may be provided by

where S is the unit Sun vector and tpH is the angle between the Sun vector and the horizon vector. Alternatively, if an iterative procedure is used to find the spacecraft attitude, the attitude vector, A, combined with the knowledge of the sensor mounting angle, y, generates the surface of a cone defined by

The horizon-in and -out vectors are obtained by simultaneously solving Eqs. (4-16), (4-20) or (4-21), and (4-26) or (4-27). For a slit horizon sensor, Eqs. (4-26) and (4-27) are replaced by

respectively, where N is defined by Eq. (4-17), ipN is the angle between the Sun vector and the slit plane normal vector at the horizon crossing, and 0 is the rotation angle between a spacecraft body meridian and the plane of the horizon sensor slit. The upper and lower signs on the right hand side of Eqs. (4-28) and (4-29) are for the horizon-in and -out crossings, respectively. In general, these simultaneous equations cannot be solved analytically and numerical methods are needed.

For example, a linear, iterative method can be used to solve Eqs. (4-16), (4-21), and (4-27). Equation (4-27) can be written as a\x + a7.y + a3z = H cos y + a, u + a2v + a3w (4-30)

where

H «[ x2+/ + z2-2( ux + vy + wz) + u2 + u2 + h-2]1/2

is the length of the horizon vector, H, and a„ a2, and a3 are the rectangular components of A. Assuming first that H is a constant, then we have two linear and one quadratic equation and (x,y,z) can be determined analytically by expressing them in the form

where i,j, and k can be 1, 2. and 3 and (xvx2.x-^ = {x,y,z).

Equations (4-31) through (4-33) can then be used to do the iteration. A good initial estimate of x^s can be found by assuming a spherical Earth and calculating x's using the above three equations with

The Effect of Earth Oblateness on Attitude Sensor Data and Solutions. One of the fundamental types of attitude measurements is the rotation angle measurement. such as the Sun-to-Earth horizon crossing or the Earth width (i.e., the rotation angle between two horizon crossings). Clearly, the time and location of the horizon triggering depend on the shape of the Earth as seen from the spacecraft. The effect of the oblateness of the Earth on the rotation angle measurements is a function of the spacecraft position and attitude and the sensor mounting angle.

Table 4-4 gives the difference in Earth width and nadir angles as computed for spheroidal and spherical Earth models. Therefore, this difference is approximately the error that would result from using a spherical Earth to model horizon sensor measurements. In these examples, the effect of oblateness tends to be greater when the spacecraft is at higher geocentric latitudes. However, with the same latitude and sensor mounting angle, the effect is not necessarily smaller when the spacecraft is

Table 4-4. Error in Nadir Angle (Atj) and Earth Width (4W) in Degrees Due to Unmodeled Oblateness. Based on circular orbit and attitude at orbit normal.

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