An application of this theorem to Eq. (12-72a) with a*,

implies that, for m large, the errors in xt, that is, Sx„ are approximately Gaussian with standard deviation

11/2

HfilH

In practice, the Central Limit Theorem may give reasonable estimates for m as small as 4 or 5, although in such cases the results should .be verified by other means. The Central Limit Theorem may also be used to compute the variance and distribution of errors in a measurement which is contaminated by many error sources with (presumably) known variances. To determine the covariance matrix Pe from Eq. (12-74), we need to determine both the measurement covariance matrix, P„, and the matrix of partial derivatives, H. In practice, Pm is normally assumed to be diagonal; i.e., the measurements are assumed to be uncorrelated. The diagonal elements of Pm are simply the variance of the measurements. If the measurements are correlated but can be written as functions of uncorrelated quantities, then the above analysis may be used to determine Pm. For example, the Sun/Earth-in and Sun/Earth-out rotation angle measurements, Q, and 80, described in Section 7.2, are correlated but may be written in terms of the uncorrected quantities, /,, the Sun to Earth-in time; tQ, the Sun to Earth-out time; and is, the Sun time as where u is the spin rate. Thus,

where a,, a,, and o, are the standard deviations in the (assumed) uncorrected v v measurements ls, l{, and tQ. The elements of H are an,

Substitution of Eqs. (12-79) into Eq. (12-78) yields

as, Tto

The correlation coefficient between the errors in il, and i20 is ol

Given an estimate of Pm, it remains to evaluate the partial derivatives 9x,/ dyj to obtain H. These partials may be computed either numerically or analytically. Numerical evaluation of partial derivatives is particularly convenient when computer evaluation of the necessary functions is already required for attitude determination, as is normally the case. For example, consider the right ascension of the spin axis attitude, a, as a function of the Sun angle, /?; a horizon sensor mounting angle, y; and other variables [Shear, 1973]: a = a(/3,y,...). Then, if the variance in fi ts t^ = and if a is linear over the appropriate range, the partial derivative is approximately da

Given specific values of p, y, and the other measurements and their variance and correlation, H can be calculated directly from Eq. (12-81).

This method breaks down if the attitude cannot be computed from the perturbed data, i.e., if a(P + afi,y,...) is undefined. (This is clearly an indication that a is nonlinear in this region and therefore Eq. (12-72) is probably invalid.) It is also possible for the perturbed solution to yield only the wrong attitude solution of an ambiguous pair and, therefore, to give absurdly large uncertainties. Numerically, both problems may be resolved by substituting some reduced fraction, e.g., 0.1 Op, for Op in Eq. (12-81). The reduced fraction chosen should be small enough to avoid undefined solutions and large enough so that computer round-off error is insignificant

The alternative to numerical partial derivatives is to find analytic expressions for the partial derivatives. (See, for example, Shear and Smith [1976] for analytic solutions for the partial derivatives for all of the spin-axis methods described in Section 11.1.) This eliminates the major inaccuracy of the numerical computations resulting either from no solution or from an erroneous one. The use of analytic partial derivatives eliminates the problem of undefined solutions or incorrect solution choice at the cost of potentially very complex algebra. The principal advantage of the numerical procedure is that it is simple and direct. In this case, the possibility of algebraic errors in the uncertainties is nearly eliminated because any error in the basic formulas will affect both the perturbed and the unperturbed solutions. Because no additional algebra is required, numerical evaluation of partials can be applied to very complex systems with minimal difficulty.

Interpretation of the Covariance Matrix. The geometrical interpretation of the computed nXn covariance matrix is generally difficult. As discussed in Section 11.3, no single number adequately represents the "attitude error" nor does the computed variance in each component of y completely characterize the "error" in that component Thus, combining attitude solutions obtained by various methods into an "average" solution by weighting according to their variance is frequently misleading.

In practice, the selected set of measurements frequently have uncorrelated errors. Thus, the measurement covariance matrix, Pm, is diagonal, and the diagonal elements of Pc, i.e., the variance in the error of the computed quantities, is given by the simple expression

However, even when Pm is diagonal, Pc is diagonal only if there is nearly a 1-to-l relationship between the measurements and the computed quantities (i.e., if Tor each x, there is a y^ such that |3x,/3y4|«|9jc,/9/y| for all k ? j. This latter condition is rarely satisfied in practice.

Further insight into the significance of the covariance matrix may be obtained by observing that it is positive definite and symmetric by construction and, if det (Pc) 0,* it may be diagonalized by a similarity transformation (see Appendix Q. This transformation may be thought of as a rotation of the n correlated errors, 5x, into a new coordinate system where the transformed errors, fix', are uncorrelated. The covariance matrix for Sx' is

p;=btpcb

*Det (/*,.)=0 if and only if |Cj| = I for some i,j. In this case, the phase space of x should be reduced by one dimension such that either x, or Xj is eliminated.

where is diagonal with elements v'x and B is the n X n matrix which diagonalizes Pc. Procedures for computing B are contained in Appendix C.

For n = 3, where the elements of x are the attitude angles, the probability that the transformed attitude error, Sx' — iSx'^Sx'^Sx^)7, is contained in the error ellipsoid

is the probability that the chi-square random variable for 3 degrees of freedom is less than K2 [Abramowitz and Stegun, 1964]. K is commonly called the a uncertainty level. Thus, the "3a" attitude error ellipsoid is defined by Eq. (12-84) with K = 3. The largest of the v', v'^, is the three-dimensional analog of the semimajor axis of the error ellipse described in Section 11.3 and is one measure of the attitude accuracy. An alternative measure of the overall attitude error when none of the are much smaller than the others is the radius, p, of a sphere whose volume equals that of the error ellipsoid. Thus,

which may be solved for p to give (see Appendix Q

P=/:(o;«;it);j),/6=x(detp;),/6=/i(detPf)1/6 (o;»«;««;,) (12-86)

Table 11-1 gives the relationship between K and various confidence levels. As an example, if we wish to assign a 99% confidence level to a three-axis attitude estimate, we obtain from Table 11-1 that K = 3.37 and use Eq. (12-83) to determine vmax- A conservative measure of the attitude accuracy is then 3.37 v'^. Alternatively, if the approximation of Eq. (12-86) is valid, we compute the determinent of the covariance matrix and set p = 3.37 [det( /><r)]1/6.

References

1. Abramowitz, M., and 1. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, No. 55, June 1964.

2. Bevington, Philip R., Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill Book Company, 1969.

3. Bryson, Arthur E., Jr., and Yu-Chi Ho, Applied Optimal Control. Waltham, MA: Ginn and Company, 1969.

4. Cayley, A., "On the Motion of Rotation of a Solid Body," Cambridge Mathematical Journal, Vol. Ill, No. 1843, p. 224-232. Reprinted in The Collected Papers of Arthur Cayley, Vol. I, Cambridge, Cambridge University Press, 1899, p. 28-35.

5. Davenport, P., Private Communication, 1968.

6. Gibbs, J. Willard, and E. B. Wilson, Vector Analysis, ed. by E. B. Wilson. New York: Scribner, 1901.

7. Goldstein, Herbert, Classical Mechanics. Reading, MA: Addison-Wesley Publishing Company, Inc., 1950.

8. Hildebrand, Francis B., Advanced Calculus for Applications. Englewood Cliffs, NJ: Prentice-Hall, Inc._1964.

9. 1MSL, Library I Reference Manual, International Mathematical and Statistical Libraries, Inc. (1MSL), Nov. 1975.

10. Keat, J., Analysis of Least-Squares Attitude Determination Routine DOAOP, Comp. Sc. Corp., CSC/TM-77/6034, Feb. 1977.

11. Klumpp, A. R., "Singularity-free Extraction of a Quaternion from a Direction Cosine Matrix," J. Spacecraft, Vol. 13, p. 754-755, 1976.

12. Korn, Granino A., and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, Inc., 1968.

13. Sabroff, A, R. Farrenkopf, A. Frew, and M. Gran, Investigation of the Acquisition Problem in Satellite Attitude Control, TRW Report AFFDL-TR-65-115, Dec. 1965.

14. Shear, M. A., Optical Aspect Attitude Determination System (OASYS), Version 4.0, System Description and Operating Guide, Comp. Sc. Corp. 3000-06000-03TM, Dec. 1973.

15. Shear, M. A., and P. M. Smith, Infrared (IR) Attitude Determination System Communications Technology Satellite Version (CTSADS) Analytical Techniques, Comp. Sc. Corp., CSC/TM-76/6149, June 1976.

16. Shuster, M. D., Algorithms for Determining Optimal Attitude Solutions, Comp. Sc. Corp., CSC/TM-78/6056, April 1978a.

17. -, Approximate Algorithms for Fast Optimal Attitude Computation, paper no. 78-1249, AIAA Guidance and Control Specialist Conference, Palo Alto, CA, Aug. 1978b.

18. Wahba, Grace, "A Least Squares Estimate of Satellite Attitude, Problem 65.1," SI AM Review, p. 384-386, Jyly 1966.

19. Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edition. Cambridge: Cambridge University Press, 1937.

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