Analysis of Flight Technical Error on Straight, Final Approach Segments

B.S. Levy, P. Som, R. Greenhaw

Abstract:	The Flight Procedure Standards Branch (AFS-420) of the Federal Aviation Administration (FAA) is responsible for developing the criteria to apply to procedures designed and implemented in the National Airspace System (NAS). With the introduction of the Required Navigation Performance (RNP) capability of modern aircraft, development of criteria for the containment widths, or minimum amount of protected airspace needed, requires the accurate statistical analysis of the magnitude of position error for these RNP systems. The position error is represented by the Total System Error (TSE) which is a combination of the Flight Technical Error (FTE) and the Navigation System Error (NSE). The NSE is the error in position due to navigation such as Global Positioning System (GPS), Distance Measuring Equipment (DME)/DME, or Very High Frequency (VHF) Omnidirectional Range (VOR)/DME. FTE is the difference between the position estimated by the Flight Management System (FMS) and the desired aircraft position. The magnitude of these errors depend upon whether the aircraft is turning, changing speed, flying straight and level, the autopilot mode (e.g., engaged) and navigation mode (e.g., lateral and/or vertical). This paper focuses only on the statistical analysis of FTE for aircraft flying straight, final approach segments. One of the challenges to the statistical analysis of FTE is limited availability of empirical data that characterizes flights under appropriate conditions. The sample size can be increased by using data from along-track locations on a flight if the errors between locations are statistically independent. Consecutive observations of FTE are, however, highly correlated because the position error data are used by the FMS to control and reduce future position errors. Each flight in the data set can be sampled at intervals at which the errors are statistically independent, to form a plot of lateral versus vertical errors (a billboard). Statistical models are developed based on the billboard data with probability density functions (pdfs). The normal, three-parameter gamma, and Johnson curve pdfs were fitted to the lateral and vertical error data from the final approach segment of the approach. The decision to fit pdfs to marginal data depends on the demonstration of independence between the lateral and vertical errors. If the non-marginal data can be used, there will be more information available for fitting the pdfs. An important question is how to accurately determine whether the lateral and vertical errors are cross-correlated, given that there may be some autocorrelation in the lateral and vertical errors. The autocorrelation in the FTE reduces the number of independent degrees of freedom used in testing the strength of cross-correlation between the lateral and vertical error components of FTE. Failure to consider non-zero autocorrelation in the lateral and vertical errors causes a higher-than-expected Type I error when testing for independence between the lateral and vertical errors. It will be more likely to conclude that there is a relation between the lateral and vertical errors when in fact there is no correlation. Another consequence is that the test on the cross-correlation will have more statistical power than warranted in the detection of a false null hypothesis. Monte Carlo simulation was used to investigate the effects of non-zero autocorrelation in the lateral and vertical errors on the Pearson's r test statistic for cross-correlation. Sets of random variables of lateral error, y, and vertical error, z, were created which contain specified amounts of autocorrelation (lag-1). Critical values as a function of Type I error, sample size, and ry and rz (i.e., the population lag-1 autocorrelation in the lateral and vertical errors, respectively) were generated that have the correct Type I error. Accurate polynomial estimating equations for the critical values were developed. Multivariate, lag-1 Markov equations for the generation of random variables containing cross-correlation and lag-1 autocorrelation were derived such that the random variation is correctly partitioned between cross-correlation and autocorrelation. These equations were used to assess the power of the Pearson's r test for cross-correlation between autocorrelated lateral and vertical errors. The multivariate equations are applied to lateral and vertical error and can be used to simulate FTE on complicated routes (e.g., curved, descending). For example, an airplane might be "high" and "right of course" on a descending, curved section of a route. Finally, the accuracy of the algorithm for selecting the Johnson curves for pdf-fitting was evaluated. Johnson curves are four-parameter pdfs that are often used to characterize skewed, tail-heavy data such as FTE.
Published in:	Proceedings of the 59th Annual Meeting of The Institute of Navigation and CIGTF 22nd Guidance Test Symposium (2003) June 23 - 25, 2003 Hyatt Regency Hotel Albuquerque, NM
Pages:	456 - 467
Cite this article:	Levy, B.S., Som, P., Greenhaw, R., "Analysis of Flight Technical Error on Straight, Final Approach Segments," Proceedings of the 59th Annual Meeting of The Institute of Navigation and CIGTF 22nd Guidance Test Symposium (2003), Albuquerque, NM, June 2003, pp. 456-467.
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