In practice, the matrix A is ill conditioned (i.e., difficult to invert in practice) for a power series, gk(x)= xk, and power series representations are not practicable for r>4 because of the greatly varying magnitude of the elements of A. However, an alternative representation using Chebyshev polynomials will greatly improve the condition of the matrix A for most applications. Chebyshev polynomials are solutions to the differential equation

/, d2& d& , . 2 „ (1 -r)—- — x—— +k îl=0 dx2 dx 6k and satisfy the recursion relation with the starting polynomials

SoW=I

Subroutines are available to set up (APCH) and solve (APFS) the normal equations, Eq. (9-8), using the Chebyshev polynomials [IBM, 1968]. APFS selects the polynomial degree by computing Sr until the equation

is satisfied, where c is an input parameter. Note that e must be greater than approximately 10~6 for single-precision arithmetic on IBM System/360 computers. Given the coefficients, Ck, the smoothed value of y, is

A residual edit may be performed by discarding data, yiy for which

where n„ is a tolerance parameter. The data are processed iteratively, first obtaining the coefficients by solving Eq; (9-8), then editing using Eq. (9-16) until no additional data are discarded and the process converges. Convergence requires a high ratio of valid to invalid data, typically 10 to 1 or greater. If the data are very noisy, or substantial data dropout is present, automatic processing will reject all data and manual intervention will be required. (See, for example, Fig. 9-2.) Note that the use of Eq. (9-16) for data validation does not depend on the method used to obtain y, and, consequently, the preceding caveat applies to any validation algorithm employing data smoothing.

An estimate of the derivative, dy,/Ax, is obtained by differentiation of Eq. (9-15), dfc d&(x)

The derivative of the Chebyshev polynomials satisfies the recursion relation [Abramowitz and Stegun 1964]

for |x| < 1 and ¿¿(± 1) equals k2 for k odd and ± k2 for k even.

Figures 9-3 through 9-5 illustrate the use of Chebyshev polynomials for data smoothing. Figure 9-3 shows GEOS-3 magnetometer data for an early orbit. Despite the highly nonlinear data, a 20th-degree Chebyshev polynomial produces a satisfactory qualitative fit, which is useful for display, for determining crude attitude rates, and for data validation. The quantitative fit is poor because the telemetered data rate is too low relative to the attitude rate; therefore, a higher degree or nonlinear representation should be used. Figure 9-4 illustrates the use of low-degree Chebyshev polynomials to fit deterministic attitude solutions. The noise on the solutions is dominated by sensor resolution and the Sun-magnetic field

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Fig. 9-3. Curve Fitting tor GEOS-3 Magnetometer Data Using a Twentieth-Degree Chebyshev Polynomial

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Fig. 9-3. Curve Fitting tor GEOS-3 Magnetometer Data Using a Twentieth-Degree Chebyshev Polynomial

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