Two-Line Element Error Analysis for MEO and GEO Satellites

Dani Racelis and Mathieu Joerger daniracelis@email.arizona.edu University of Arizona

2018 PNT Symposium

2018 November 7, Wednesday

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1800 functioning

satellites

Photo credits: ESA; Numbers from ESA’s Space Debris Office at ESOC, Darmstadt Germany, “Space Debris by the Numbers” (valid as of January 2018), and Nature article “The quest to conquer Earth’s space junk problem”

21 warnings of potential space

collisions

21,000 debris objects regularly tracked

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Motivation

•  Major Challenge in space traffic management (STM): Quantifying and predicting collision risk and false alert risk

•  The U.S. TLE database is the most comprehensive catalogue of space objects readily accessible to the public.

[The Aerospace Corporation (2015)]

The Orbits We Use – Tyler Reid

[Aero] The Aerospace Corporation. “Understanding Space Debris Causes, Mitigations, and Issues.” Crosslink Magazine. (2015). [Vallado] Vallado, D.A., and P.J. Cefola. Two-Line Element Sets - Practice and Use. Paper IAC-12.A6.6.11 63rd IAC. 1-2 October 2012. Naples, Italy. [Reid] Reid, Tyler G.R., Neish, Andrew M., Walter, Todd, Enge, Per K., "Broadband LEO Constellations for Navigation", NAVIGATION, 65(2), 2018.

[Vallado (2012)]

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Prior Work

•  TLEs are used in various applications including –  conjunction assessment [Flohrer CA] –  space object association [Weigel] –  improving orbit determination accuracy [Flohrer]

•  Prior work often focus on TLE propagation errors using simplified general perturbation models, which focus on how the positioning error grows when the TLE is propagated from the reported TLE epoch. [Kelso, Vallado]

[Flohrer CA] Flohrer, T., H. Krag & H. Klinkrad (2009). ESA’s Process for the Identification and Assessment of High-risk Conjunction Events. Adv. Sp. Res. [Flohrer] Flohrer, T., H. Krag, H. Klinkrad, B. Bastida Virgili, and C. Frϋh. 2009. Improving ESA's Collision Risk Estimates by an Assessment of the TLE Orbit

Errors of the US SSN Catalogue. Paper S4.4 5th European Space Debris Conference. (2009) [Weigel] Weigel, M., M. Meinel, H. Fiedler. "Processing of Optical Telescope Obserations with the Space Object Catalogue BACARDI." ISSFD. (2015). [Kelso] Kelso, T.S., 2007 Validation of SGP4 and ICD-GPS-200 against GPS precision Ephemerides. AAS 07-127 AAS/AIAA. (2007). [Vallado] Vallado, D.A., and P.J. Cefola. Two-Line Element Sets - Practice and Use. IAC-12.A6.6.11 63rd IAC. (2012).

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Focus and Contributions

Our focus is on quantifying TLE positioning error at the reported TLE epoch, for hundreds of thousands of data points, and building a model for it.

–  We analyze spacecraft at known locations, i.e. GPS and WAAS.

1.  We processed 20 years of TLE data to evaluate TLE errors for GPS and WAAS satellites.

2.  We derived probabilistic bounds on space object positioning errors using overbounding theory.

–  We leverage prior analytical methods in aviation navigation safety [DeCleene, Rife]

[DeCleene] DeCleene, B. “Defining Pseudorange Integrity – Overbounding.” Proc. of ION GPS 2000. Salt Lake City, UT. (2000). [Rife] Rife, Pullen, Enge, Pervan. “Paired Overbounding for Nonideal LAAS and WAAS Error Distributions.” IEEE TAES. 42(4). (2006).

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Data Acquisition

Two-Line Element Sets

Estimated position and velocity

GPS broadcast message

‘True’ position and velocity

Space-Track CDDIS Database

Simplified General Perturbations

SGP4

17-parameter model

Spacecraft Positioning

(Propagators)

Data Format

Sources of Satellite

Orbital Data

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Process Overview

GPS broadcast message

Two-Line Element Sets

‘True’ position and velocity

Estimated position and velocity

•  Available at two-hour intervals •  Retrieved separately for each TLE •  GPS time (leap seconds)

Synchronized

Converted using SGP4/SDP4

Propagated to TLE epoch

At this point I will

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Process Overview

GPS broadcast message

Two-Line Element Sets

‘True’ position and velocity

Estimated position and velocity

3D Error Distribution

Error Model

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adfs

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TLE-Based Positioning Errors for MEO Satellites (GPS)

148,984 data points for 30 satellites from July 23 1997 to March 31 2018

– outlier analysis: in [ paper ] –

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Abrupt Changes in TLE Errors

asdf

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Abrupt Changes in TLE Errors

asdf

October 21, 2015

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TLE Errors on MEO (GPS) and GEO (WAAS)

asdf

October 21, 2015

WAAS data analysis in [paper]

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TLE Error Statistics Over Time

TLE error mean and standard deviation reach constant values.

The ‘stationary’ period: October 21, 2015 to March 31, 2018 is used for statistical analysis, and includes 47,626 data points

October 21, 2015

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TLE Error Statistics Over 30 GPS Satellites

Period of constant standard deviation values is observed on all GPS satellites

in all directions (along-track, cross-track, and radial)

Color code identifies different satellites

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Sample Statistics are Insufficient

• 

[FAA] Federal Aviation Administration (FAA). “System Design and Analysis.” Advisory Circular AC 25.1309-1A. (1988).

• 

N > 10,000 points

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Probabilistic Bound on TLE Errors

[DeCleene] DeCleene, “Defining Pseudorange Integrity - Overbounding,” IONGPS 2000. [Rife] Rife et al., “Paired Overbounding for Nonideal LAAS and WAAS Error Distributions.” IEEE TAES, 2006. [Blanch] Blanch, Juan, Walter, Todd, Enge, Per, "A MATLAB Toolset to Determine Strict Gaussian Bounding Distributions of a Sample Distribution," 30th ITM

of The Satellite Division of the Institute of Navigation (ION GNSS+ 2017), Portland, Oregon, September 2017, pp. 4236-4247.

•  We use methods in aviation navigation safety to define probabilistic bounds on wide-tailed non-Gaussian TLE errors. [DeCleene, Rife, Blanch]

•  Using overbounding theory, probabilistic bounds on the entire positioning error distribution can be achieved.

–  provides the means to model empirical error distributions using familiar parametric models, e.g., Gaussian distributions.

•  Tight bounds are needed: –  the model must be conservative to be safe, but should not be

overly conservative or there would be too many collision warnings

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Paired Overbounding

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Use the phrase “paired overbounding.” Be careful because there are different

kinds of overbounding. • 

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Paired Overbounding

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Paired Overbounding

GPS TLE along-track errors can be modeled using a pair of overbounding Gaussian functions with

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Conclusions

•  We quantified TLE-based positioning errors for GPS satellites using 20 years of data (148,984 data points)

–  GPS satellites serve as surrogate for space objects to evaluate space debris tracking performance

•  A probabilistic bound on TLE-based positioning errors was determined

–  We will use similar methods to characterize telescope error data

•  In the next steps of this work, we will use actual observations to develop observation error models which can be used to evaluate collision risks