# Satellite Orbit by gjjur4356

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```									Performance Evaluation
of Several
Interpolation Methods
for GPS Satellite Orbit

Supervised by Dr. Ahmed El-Rabbany
Presentation Topics

Introduction
Errors of Interpolation

Lagrange Method

Newton Divided Difference Method

Trigonometric Method

Conclusion
Introduction
The IGS have developed three precise GPS
ephemerides:
Ultra rapid
Rapid

Final
These ephemerides are spaced at 15 minutes
intervals but many GPS applications require
precise ephemeris at higher rates, which is
the reason for interpolation.
Interpolation Errors
Function Related Error:
The amount of this error can be used as a
measure of how well the interpolating
method approaches the actual value of the
time series.
Computer Generated Error:

This error is the result of computer
limitations. It depends on the operating
system, programming language and more or
less on computer hardware.
Interpolation Properties
   Taking too few points produces an
unreliable interpolation output.
   Taking a plenty of points is ideally
convenient. However, the computer
capability is limited up to a specific
number of points beyond which the
computer behaves unpredictably.
near the end points and tends to
improve as the interpolator moves
towards the center.
Lagrange Method

•Lagrange Formula:
Lagrange Interpolation

•INTERPOLATION ALGORITHM:

The 24-hour data is divided into 23 overlapping segments each
of 9 terms as shown below:

SEGMENT 1   SEGMENT 2                                SEGMENT 22   SEGMENT 23

00:00                                                                  23:45
Newton Divided Difference Interpolation
•Newton Divided Difference Formula:
Trigonometric Interpolation
•The Trigonometric Series:
•This method is suggested by [Mark
Schenewerk, A brief review of basic GPS
orbit interpolation strategies, 2002].
•The code is taken from:
http://www.noaa.gov/gps-toolbox/sp3intrp
•The Trigonometric coefficients are computed
using an algorithm called Singular Value
Decomposition.
Comparison between
Lagrange and Trigonometric* Interpolation

MEAN (cm)                       STD (cm)                     MAX (cm)
INERTIAL ORBIT
dx        dy          dz        dx        dy        dz       dx        dy        dz

TRIGONOMETRIC     0.0010    O.OO35      0.0007    0.0499    0.0841    0.0654   0.3000    0.5000    0.4000

LAGRANGE       0.0025    0.0067      0.0037    0.0451    0.0756    0.0405   0.4127    0.6374    0.2233

MEAN (cm)                       STD (cm)                     MAX (cm)
ECEF ORBIT
dx        dy         dz        dx        dy         dz      dx         dy        dz

TRIGONOMETRIC     0.0126    0.0007     0.0021    0.1032    0.0580     0.0696    1.2      0.3000    0.2000

LAGRANGE        0.0016   0.0120      0.0034    0.1548    0.2501     0.0623   1.5216    3.3753    0.4276

* The boundaries of the Trigonometric are not included. According to Schenewerk
(2003) the error at the boundaries is 8.2 cm for INERTIAL and 10.3 cm for ECEF.
The direct interpolation of IGS precise
ephemeris has one drawback. The very high
positive and very low negative values (km)
make it difficult to get an accuracy of
millimeter level. As another alternative we
precise ephemeris whose values are in
meters and therefore it would be easier to
get millimeter accuracy.
Conclusion
   Lagrange and Newton Divided Difference demonstrate
completely identical results in terms of interpolation error.
   Excluding the boundaries, the Trigonometric method
yielded the best accuracy of all interpolation methods due
to the periodic nature of the GPS orbit. This problem can
be avoided by centering the day to be interpolated among
sufficient data before and after the day. However, in real
time applications no data can be added after the day.
   Lagrange has a better performance at the boundaries
which makes it more convenient for real time
applications.
    The interpolation via the broadcast ephemeris has
produced the best results within the two-hour ephemeris
period.
References

Press, W.H., S.A. Teukolosky, W.T. Vetterling, B.P.
Flannery (2002). Numerical Recipes in C++: The
Art of Scientific Computing. Cambridge
University Press.
Schenewerk, M. (2003). “A Brief Review Of Basic GPS
Orbit Interpolation Strategies.” GPS Solutions,
Vol. 6, No. 4, pp. 265-267.
Spiegel, M.R. (1999). Mathematical Handbook of
Formulas and Tables. McGraw Hill. Armed
Forced, Munich.

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