In Eq. (11-35), Um and Un are the total uncertainties in measurements m and n; Rm and R^ are the random errors in measurements m and n\ AS, is the ith systematic error existing in either measurement; dm/ dS, and dn/dS, are the partial derivatives of m and n with respect to the ith systematic error: and Cm/„ is the correlated uncertainty component between the two measurements.

In this case, the attitude uncertainty can be obtained from the covariance matrix approach given in Section II.3.1 with Eq. (II-25) replaced by Eq. (11-34). The result is


Equation (11-36) gives the general expression for the attitude uncertainty determined by two measurements with total uncertainties Um and Un, and correlated uncertainty component Cm/n. This equation can be applied to any single-axis attitude determination procedure regardless of the type of measurements and attitude determination methods.

Equation (11-36) shows that the attitude accuracy in general is determined by three factors: the measurement uncertainties Um, Un, and Cm/n; the measurement densities, dm and dn\ and the correlation angle, 0m/„- Note that the attitude uncertainty goes to infinity (i.e., a singularity occurs) whenever dm, d„, or sin ®m/„ is zero.

The expressions for the measurement uncertainties are given in Eq. (11-35), and the expressions for d and 0, which depend on the types of the two measurements, are given in Section 11.3.3 for arc-length and rotation angle measurements.

1133 Measurement Densities and Correlation Angles

Expressions for the measurement density, dm, and the correlation angle, 0m/„, depend on the types of measurements. Because arc-length and rotation angles are the most fundamental and most commonly used measurements, we derive explicit expressions for dm, ®m/„, and UA in terms of the geometrical parameters involved. The results are presented in Table 11-2 using the notation defined in Figs. 11-11 and 11-14. The attitude uncertainty UA, for any deterministic attitude method using arc-length and rotation angle measurements, can be obtained by substituting the expressions from Table 11-2 into Eq. (11-31) or (11-36).

To make the discussion specific, the Sun and the Earth are used as the two reference vectors. However, final expressions are not limited to the Sun/Earth system. The results are generally applicable for any single-axis attitude determination procedure using arc-length or rotation angle measurements. We emphasize that the uncertainties presented in Table 11-2 are a result of the observations which are used for a deterministic solution and do not depend on the numerical procedure by which the attitude is computed. For example, Section' describes a procedure for computing the attitude from the measurements /3 and First, ft, $ and the reference vector parameters are used to compute ij, and then /3 and ij are used to compute the attitude. The uncertainty for this method may be obtained directly from line 3 of Table 11-2, irrespective of the fact that 17 was used as a numerically convenient intermediate variable in computing the attitude.

Table 11-2 gives the attitude uncertainty in terms of simple functions of measurement uncertainties and geometrical conditions, which enables one to give quick attitude uncertainty estimates, frequently without computer computations. This is a major advantage of the geometrical approach over other computational techniques, in terms of time, cost, and the need for prompt decisions.

Table 11-2. Summary of Single-Axis Attitude Accuracy for Arc-Length and Rotation Angle Measurements. (See Section 11.5 for a list of attitude singularities for these methods.)
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